GCSE Maths: Changing the subject: brackets, fractions, powers and roots (Foundation)

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 3(x + 2)
   Reading from the subject outwards, what happens to x: 2 is added; then it is multiplied by 3.
   To undo it we reverse the order: divide by 3, then subtract 2.
2. Divide both sides by 3. x is being multiplied by 3 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 3.
   y3 = 3(x + 2)3
   y3 = x + 2
3. Subtract 2 from both sides. 2 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 2 from both sides. Doing the same to both sides keeps the formula true.
   y3 − 2 = x + 2 − 2
   y3 − 2 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y3 − 2

Check. Pick numbers for the other letters: y = 9. The new formula gives
x = 93 − 2 = 3 − 2 = 1
Now put x = 1 back into the original formula. Left side: 9. Right side: 3 × (1 + 2) = 3 × 3 = 9. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 5(x − 1)
   Reading from the subject outwards, what happens to x: 1 is subtracted; then it is multiplied by 5.
   To undo it we reverse the order: divide by 5, then add 1.
2. Divide both sides by 5. x is being multiplied by 5 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 5.
   y5 = 5(x − 1)5
   y5 = x − 1
3. Add 1 to both sides. 1 is being subtracted. The inverse of subtracting is adding, so add 1 to both sides.
   y5 + 1 = x − 1 + 1
   y5 + 1 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y5 + 1

Check. Pick numbers for the other letters: y = 10. The new formula gives
x = 105 + 1 = 2 + 1 = 3
Now put x = 3 back into the original formula. Left side: 10. Right side: 5 × (3 − 1) = 5 × 2 = 10. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 4(x + 3) − 1
   Reading from the subject outwards, what happens to x: 3 is added; then it is multiplied by 4; then 1 is subtracted.
   To undo it we reverse the order: add 1, then divide by 4, then subtract 3.
2. Add 1 to both sides. 1 is being subtracted. The inverse of subtracting is adding, so add 1 to both sides.
   y + 1 = 4(x + 3) − 1 + 1
   y + 1 = 4(x + 3)
3. Divide both sides by 4. x is being multiplied by 4 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 4.
   y + 14 = 4(x + 3)4
   y + 14 = x + 3
4. Subtract 3 from both sides. 3 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 3 from both sides. Doing the same to both sides keeps the formula true.
   y + 14 − 3 = x + 3 − 3
   y + 14 − 3 = x
5. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y + 14 − 3

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 2(3x − 1)
   Reading from the subject outwards, what happens to x: it is multiplied by 3; then 1 is subtracted; then it is multiplied by 2.
   To undo it we reverse the order: divide by 2, then add 1, then divide by 3.
2. Divide both sides by 2. x is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   y2 = 2(3x − 1)2
   y2 = 3x − 1
3. Add 1 to both sides. 1 is being subtracted. The inverse of subtracting is adding, so add 1 to both sides.
   y2 + 1 = 3x − 1 + 1
   y2 + 1 = 3x
4. Divide both sides by 3. x is being multiplied by 3 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 3.
   y2 + 13 = 3x3
   y2 + 13 = x
5. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y2 + 13

Check. Pick numbers for the other letters: y = 4. The new formula gives
x = 42 + 13 = 2 + 13 = 33 = 1
Now put x = 1 back into the original formula. Left side: 4. Right side: 2 × (3 × 1 − 1) = 2 × (3 − 1) = 2 × 2 = 4. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until l is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get l on its own we must undo everything that has been done to it, in the reverse order.
   Formula: P = 2(l + w)
   Reading from the subject outwards, what happens to l: w is added; then it is multiplied by 2.
   To undo it we reverse the order: divide by 2, then subtract w.
2. Divide both sides by 2. l is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   P2 = 2(l + w)2
   P2 = l + w
3. Subtract w from both sides. w is being added to the part containing l. The inverse (opposite) of adding is subtracting, so subtract w from both sides. Doing the same to both sides keeps the formula true.
   P2 − w = l + w − w
   P2 − w = l
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   l = P2 − w

Check. Pick numbers for the other letters: P = 6, w = 2. The new formula gives
l = 62 − 2 = 3 − 2 = 1
Now put l = 1 back into the original formula. Left side: 6. Right side: 2 × (1 + 2) = 2 × 3 = 6. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until b is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get b on its own we must undo everything that has been done to it, in the reverse order.
   Formula: A = 12(a + b)h
   Reading from the subject outwards, what happens to b: a is added; then it is multiplied by h; then it is multiplied by 12.
   To undo it we reverse the order: multiply by 2, then divide by h, then subtract a.
2. Multiply both sides by 2. 12 means "divide by 2". The inverse of dividing is multiplying, so multiply both sides by 2.
   A × 2 = 12(a + b)h × 2
   2A = (a + b)h
3. Divide both sides by h. b is being multiplied by h (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by h.
   2Ah = (a + b)hh
   2Ah = a + b
4. Subtract a from both sides. a is being added to the part containing b. The inverse (opposite) of adding is subtracting, so subtract a from both sides. Doing the same to both sides keeps the formula true.
   2Ah − a = a + b − a
   2Ah − a = b
5. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   b = 2Ah − a

Check. Pick numbers for the other letters: A = 10, a = 8, h = 2. The new formula gives
b = 2 × 102 − 8 = 202 − 8 = 10 − 8 = 2
Now put b = 2 back into the original formula. Left side: 10. Right side: 12 × (8 + 2) × 2 = 12 × 10 × 2 = 5 × 2 = 10. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until C is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get C on its own we must undo everything that has been done to it, in the reverse order.
   Formula: F = 95C + 32
   Reading from the subject outwards, what happens to C: it is multiplied by 95; then 32 is added.
   To undo it we reverse the order: subtract 32, then multiply by 59.
2. Subtract 32 from both sides. 32 is being added to the part containing C. The inverse (opposite) of adding is subtracting, so subtract 32 from both sides. Doing the same to both sides keeps the formula true.
   F − 32 = 95C + 32 − 32
   F − 32 = 95C
3. Multiply both sides by 59. C is being multiplied by 95. To undo that, multiply by its reciprocal 59 (the fraction turned upside down), because 95 × 59 = 1.
   (F − 32) × 59 = 95C × 59
   59(F − 32) = C
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   C = 59(F − 32)

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until a is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get a on its own we must undo everything that has been done to it, in the reverse order.
   Formula: S = 2(ab + bc)
   Reading from the subject outwards, what happens to a: it is multiplied by b; then bc is added; then it is multiplied by 2.
   To undo it we reverse the order: divide by 2, then subtract bc, then divide by b.
2. Divide both sides by 2. a is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   S2 = 2(ab + bc)2
   S2 = ab + bc
3. Subtract bc from both sides. bc is being added to the part containing a. The inverse (opposite) of adding is subtracting, so subtract bc from both sides. Doing the same to both sides keeps the formula true.
   S2 − bc = ab + bc − bc
   S2 − bc = ab
4. Divide both sides by b. a is being multiplied by b (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by b.
   S2 − bcb = abb
   S2 − bcb = a
5. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   a = S2 − bcb

Check. Pick numbers for the other letters: S = 8, b = 1, c = 2. The new formula gives
a = 82 − 1 × 21 = 4 − 21 = 21 = 2
Now put a = 2 back into the original formula. Left side: 8. Right side: 2 × (2 × 1 + 1 × 2) = 2 × (2 + 2) = 2 × 4 = 8. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = x5
   Reading from the subject outwards, what happens to x: it is divided by 5.
   To undo it we reverse the order: multiply by 5.
2. Multiply both sides by 5. The part containing x is being divided by 5. The inverse of dividing is multiplying, so multiply both sides by 5.
   y × 5 = x5 × 5
   5y = x
3. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = 5y

Check. Pick numbers for the other letters: y = 6. The new formula gives
x = 5 × 6 = 30
Now put x = 30 back into the original formula. Left side: 6. Right side: 305 = 6. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = x + 32
   Reading from the subject outwards, what happens to x: 3 is added; then it is divided by 2.
   To undo it we reverse the order: multiply by 2, then subtract 3.
2. Multiply both sides by 2. The part containing x is being divided by 2. The inverse of dividing is multiplying, so multiply both sides by 2.
   y × 2 = x + 32 × 2
   2y = x + 3
3. Subtract 3 from both sides. 3 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 3 from both sides. Doing the same to both sides keeps the formula true.
   2y − 3 = x + 3 − 3
   2y − 3 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = 2y − 3

Check. Pick numbers for the other letters: y = 9. The new formula gives
x = 2 × 9 − 3 = 18 − 3 = 15
Now put x = 15 back into the original formula. Left side: 9. Right side: 15 + 32 = 182 = 9. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 2x − 13
   Reading from the subject outwards, what happens to x: it is multiplied by 2; then 1 is subtracted; then it is divided by 3.
   To undo it we reverse the order: multiply by 3, then add 1, then divide by 2.
2. Multiply both sides by 3. The part containing x is being divided by 3. The inverse of dividing is multiplying, so multiply both sides by 3.
   y × 3 = 2x − 13 × 3
   3y = 2x − 1
3. Add 1 to both sides. 1 is being subtracted. The inverse of subtracting is adding, so add 1 to both sides.
   3y + 1 = 2x − 1 + 1
   3y + 1 = 2x
4. Divide both sides by 2. x is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   3y + 12 = 2x2
   3y + 12 = x
5. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = 3y + 12

Check. Pick numbers for the other letters: y = 3. The new formula gives
x = 3 × 3 + 12 = 9 + 12 = 102 = 5
Now put x = 5 back into the original formula. Left side: 3. Right side: 2 × 5 − 13 = 10 − 13 = 93 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: M = x + y2
   Reading from the subject outwards, what happens to x: y is added; then it is divided by 2.
   To undo it we reverse the order: multiply by 2, then subtract y.
2. Multiply both sides by 2. The part containing x is being divided by 2. The inverse of dividing is multiplying, so multiply both sides by 2.
   M × 2 = x + y2 × 2
   2M = x + y
3. Subtract y from both sides. y is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract y from both sides. Doing the same to both sides keeps the formula true.
   2M − y = x + y − y
   2M − y = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = 2M − y

Check. Pick numbers for the other letters: M = 10, y = 6. The new formula gives
x = 2 × 10 − 6 = 20 − 6 = 14
Now put x = 14 back into the original formula. Left side: 10. Right side: 14 + 62 = 202 = 10. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until s is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get s on its own we must undo everything that has been done to it, in the reverse order.
   Formula: A = s²
   Reading from the subject outwards, what happens to s: it is squared.
   To undo it we reverse the order: take the square root.
2. Take the square root of both sides. s is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √A = √s²
   √A = s
3. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   s = √A

Check. Pick numbers for the other letters: A = 4. The new formula gives
s = √4 = 2
Now put s = 2 back into the original formula. Left side: 4. Right side: 2² = 4. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until r is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get r on its own we must undo everything that has been done to it, in the reverse order.
   Formula: A = πr²
   Reading from the subject outwards, what happens to r: it is squared; then it is multiplied by π.
   To undo it we reverse the order: divide by π, then take the square root.
2. Divide both sides by π. r is being multiplied by π (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by π.
   Aπ = πr²π
   Aπ = r²
3. Take the square root of both sides. r is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √Aπ = √r²
   √Aπ = r
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   r = √Aπ

Check. Pick numbers for the other letters (for this check only, let π = 3 to keep the numbers easy): A = 3, π = 3. The new formula gives
r = √33 = √1 = 1
Now put r = 1 back into the original formula. Left side: 3. Right side: 3 × 1² = 3 × 1 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = x² + 3
   Reading from the subject outwards, what happens to x: it is squared; then 3 is added.
   To undo it we reverse the order: subtract 3, then take the square root.
2. Subtract 3 from both sides. 3 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 3 from both sides. Doing the same to both sides keeps the formula true.
   y − 3 = x² + 3 − 3
   y − 3 = x²
3. Take the square root of both sides. x is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √y − 3 = √x²
   √y − 3 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = √y − 3

Check. Pick numbers for the other letters: y = 4. The new formula gives
x = √4 − 3 = √1 = 1
Now put x = 1 back into the original formula. Left side: 4. Right side: 1² + 3 = 1 + 3 = 4. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until v is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get v on its own we must undo everything that has been done to it, in the reverse order.
   Formula: E = 12mv²
   Reading from the subject outwards, what happens to v: it is squared; then it is multiplied by m; then it is multiplied by 12.
   To undo it we reverse the order: multiply by 2, then divide by m, then take the square root.
2. Multiply both sides by 2. 12 means "divide by 2". The inverse of dividing is multiplying, so multiply both sides by 2.
   E × 2 = 12mv² × 2
   2E = mv²
3. Divide both sides by m. v is being multiplied by m (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by m.
   2Em = mv²m
   2Em = v²
4. Take the square root of both sides. v is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √2Em = √v²
   √2Em = v
5. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   v = √2Em

Check. Pick numbers for the other letters: E = 10, m = 5. The new formula gives
v = √2 × 105 = √205 = √4 = 2
Now put v = 2 back into the original formula. Left side: 10. Right side: 12 × 5 × 2² = 12 × 5 × 4 = 2.5 × 4 = 10. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 2x²
   Reading from the subject outwards, what happens to x: it is squared; then it is multiplied by 2.
   To undo it we reverse the order: divide by 2, then take the square root.
2. Divide both sides by 2. x is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   y2 = 2x²2
   y2 = x²
3. Take the square root of both sides. x is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √y2 = √x²
   √y2 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = √y2

Check. Pick numbers for the other letters: y = 8. The new formula gives
x = √82 = √4 = 2
Now put x = 2 back into the original formula. Left side: 8. Right side: 2 × 2² = 2 × 4 = 8. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = √x
   Reading from the subject outwards, what happens to x: the square root is taken.
   To undo it we reverse the order: square.
2. Square both sides. x is under a square root. The inverse of taking a root is squaring, so do that to both sides.
   y² = (√x)²
   y² = x
3. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y²

Check. Pick numbers for the other letters: y = 5. The new formula gives
x = 5² = 25
Now put x = 25 back into the original formula. Left side: 5. Right side: √25 = 5. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = √x + 3
   Reading from the subject outwards, what happens to x: the square root is taken; then 3 is added.
   To undo it we reverse the order: subtract 3, then square.
2. Subtract 3 from both sides. 3 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 3 from both sides. Doing the same to both sides keeps the formula true.
   y − 3 = √x + 3 − 3
   y − 3 = √x
3. Square both sides. x is under a square root. The inverse of taking a root is squaring, so do that to both sides.
   (y − 3)² = (√x)²
   (y − 3)² = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = (y − 3)²

Check. Pick numbers for the other letters: y = 4. The new formula gives
x = (4 − 3)² = 1² = 1
Now put x = 1 back into the original formula. Left side: 4. Right side: √1 + 3 = 1 + 3 = 4. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until h is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get h on its own we must undo everything that has been done to it, in the reverse order.
   Formula: v = √2gh
   Reading from the subject outwards, what happens to h: it is multiplied by 2g; then the square root is taken.
   To undo it we reverse the order: square, then divide by 2g.
2. Square both sides. h is under a square root. The inverse of taking a root is squaring, so do that to both sides.
   v² = (√2gh)²
   v² = 2gh
3. Divide both sides by 2g. h is being multiplied by 2g (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2g.
   v²2g = 2gh2g
   v²2g = h
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   h = v²2g

Check. Pick numbers for the other letters: v = 4, g = 2. The new formula gives
h = 4²2 × 2 = 164 = 4
Now put h = 4 back into the original formula. Left side: 4. Right side: √2 × 2 × 4 = √4 × 4 = √16 = 4. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = √5x
   Reading from the subject outwards, what happens to x: it is multiplied by 5; then the square root is taken.
   To undo it we reverse the order: square, then divide by 5.
2. Square both sides. x is under a square root. The inverse of taking a root is squaring, so do that to both sides.
   y² = (√5x)²
   y² = 5x
3. Divide both sides by 5. x is being multiplied by 5 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 5.
   y²5 = 5x5
   y²5 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y²5

Check. Pick numbers for the other letters: y = 10. The new formula gives
x = 10²5 = 1005 = 20
Now put x = 20 back into the original formula. Left side: 10. Right side: √5 × 20 = √100 = 10. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until r is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get r on its own we must undo everything that has been done to it, in the reverse order.
   Formula: A = πr²
   Reading from the subject outwards, what happens to r: it is squared; then it is multiplied by π.
   To undo it we reverse the order: divide by π, then take the square root.
2. Divide both sides by π. r is being multiplied by π (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by π.
   Aπ = πr²π
   Aπ = r²
3. Take the square root of both sides. r is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √Aπ = √r²
   √Aπ = r
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   r = √Aπ

Check. Pick numbers for the other letters (for this check only, let π = 3 to keep the numbers easy): A = 3, π = 3. The new formula gives
r = √33 = √1 = 1
Now put r = 1 back into the original formula. Left side: 3. Right side: 3 × 1² = 3 × 1 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   r = √Aπ
2. Substitute: A = 28.26, π = 3.14. Substitute means replace each letter with its number.
   r = √28.263.14
3. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   28.263.14 means 28.26 ÷ 3.14 = 9
   r = √9
4. Work out the root. A square root asks: which number multiplied by itself gives this? The root sign works like a bracket, so the number under it is worked out first.
   √9 = 3 because 3 × 3 = 9
   r = 3
5. Write the answer with its units. The answer should say what was being worked out.
   r = 3 m

Check. Put the numbers straight into a calculator: √28.26/3.14 gives 3. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until t is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get t on its own we must undo everything that has been done to it, in the reverse order.
   Formula: d = 12gt²
   Reading from the subject outwards, what happens to t: it is squared; then it is multiplied by g; then it is multiplied by 12.
   To undo it we reverse the order: multiply by 2, then divide by g, then take the square root.
2. Multiply both sides by 2. 12 means "divide by 2". The inverse of dividing is multiplying, so multiply both sides by 2.
   d × 2 = 12gt² × 2
   2d = gt²
3. Divide both sides by g. t is being multiplied by g (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by g.
   2dg = gt²g
   2dg = t²
4. Take the square root of both sides. t is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √2dg = √t²
   √2dg = t
5. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   t = √2dg

Check. Pick numbers for the other letters: d = 8, g = 4. The new formula gives
t = √2 × 84 = √164 = √4 = 2
Now put t = 2 back into the original formula. Left side: 8. Right side: 12 × 4 × 2² = 12 × 4 × 4 = 2 × 4 = 8. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   t = √2dg
2. Substitute: d = 80, g = 10. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   t = √2 × 8010
3. Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.
   2 × 80 = 160
   t = √16010
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   16010 means 160 ÷ 10 = 16
   t = √16
5. Work out the root. A square root asks: which number multiplied by itself gives this? The root sign works like a bracket, so the number under it is worked out first.
   √16 = 4 because 4 × 4 = 16
   t = 4
6. Write the answer with its units. The answer should say what was being worked out.
   t = 4 seconds

Check. Put the numbers straight into a calculator: √(2 × 80)/10 gives 4. ✓

Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until v is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get v on its own we must undo everything that has been done to it, in the reverse order.
   Formula: E = 12mv²
   Reading from the subject outwards, what happens to v: it is squared; then it is multiplied by m; then it is multiplied by 12.
   To undo it we reverse the order: multiply by 2, then divide by m, then take the square root.
2. Multiply both sides by 2. 12 means "divide by 2". The inverse of dividing is multiplying, so multiply both sides by 2.
   E × 2 = 12mv² × 2
   2E = mv²
3. Divide both sides by m. v is being multiplied by m (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by m.
   2Em = mv²m
   2Em = v²
4. Take the square root of both sides. v is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √2Em = √v²
   √2Em = v
5. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   v = √2Em

Check. Pick numbers for the other letters: E = 10, m = 5. The new formula gives
v = √2 × 105 = √205 = √4 = 2
Now put v = 2 back into the original formula. Left side: 10. Right side: 12 × 5 × 2² = 12 × 5 × 4 = 2.5 × 4 = 10. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   v = √2Em
2. Substitute: E = 100, m = 2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   v = √2 × 1002
3. Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.
   2 × 100 = 200
   v = √2002
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   2002 means 200 ÷ 2 = 100
   v = √100
5. Work out the root. A square root asks: which number multiplied by itself gives this? The root sign works like a bracket, so the number under it is worked out first.
   √100 = 10 because 10 × 10 = 100
   v = 10
6. Write the answer with its units. The answer should say what was being worked out.
   v = 10 m/s

Check. Put the numbers straight into a calculator: √(2 × 100)/2 gives 10. ✓

Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until h is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get h on its own we must undo everything that has been done to it, in the reverse order.
   Formula: A = h(a + b)2
   Reading from the subject outwards, what happens to h: it is multiplied by (a + b); then it is divided by 2.
   To undo it we reverse the order: multiply by 2, then divide by (a + b).
2. Multiply both sides by 2. The part containing h is being divided by 2. The inverse of dividing is multiplying, so multiply both sides by 2.
   A × 2 = h(a + b)2 × 2
   2A = h(a + b)
3. Divide both sides by (a + b). h is being multiplied by (a + b) (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by (a + b).
   2A(a + b) = h(a + b)(a + b)
   2A(a + b) = h
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   h = 2A(a + b)

Check. Pick numbers for the other letters: A = 4, a = 2, b = 6. The new formula gives
h = 2 × 4(2 + 6) = 2 × 48 = 88 = 1
Now put h = 1 back into the original formula. Left side: 4. Right side: 1 × (2 + 6)2 = 1 × 82 = 82 = 4. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   h = 2Aa + b
2. Substitute: A = 48, a = 5, b = 7. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   h = 2 × 485 + 7
3. Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.
   2 × 48 = 96
   5 + 7 = 12
   h = 9612
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   9612 means 96 ÷ 12 = 8
   h = 8
5. Write the answer with its units. The answer should say what was being worked out.
   h = 8 cm

Check. Put the numbers straight into a calculator: (2 × 48)/(5 + 7) gives 8. ✓

Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 2x²
   Reading from the subject outwards, what happens to x: it is squared; then it is multiplied by 2.
   To undo it we reverse the order: divide by 2, then take the square root.
2. Divide both sides by 2. x is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   y2 = 2x²2
   y2 = x²
3. Take the square root of both sides. x is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √y2 = √x²
   √y2 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = √y2

Check. Pick numbers for the other letters: y = 8. The new formula gives
x = √82 = √4 = 2
Now put x = 2 back into the original formula. Left side: 8. Right side: 2 × 2² = 2 × 4 = 8. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   x = √y2
2. Substitute: y = 50. Substitute means replace each letter with its number.
   x = √502
3. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   502 means 50 ÷ 2 = 25
   x = √25
4. Work out the root. A square root asks: which number multiplied by itself gives this? The root sign works like a bracket, so the number under it is worked out first.
   √25 = 5 because 5 × 5 = 25
   x = 5

Check. Put the numbers straight into a calculator: √50/2 gives 5. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Method. A negative number squared is positive, so a negative number can also work.

1. Test x = −5. Put the negative number in brackets.
   y = 2 × (−5)² = 2 × 25 = 50
2. Conclude. It gives 50 as well.
   x = −5

Check. Both 5 and −5 satisfy y = 2x² = 50. ✓

Watch out. We usually take only the positive root when x is a real length, but in algebra both roots are valid.