GCSE Maths · Quadratic and Simultaneous Equations · Foundation · Summary
GCSE Maths Quadratic and Simultaneous Equations (Foundation): summary and key words
Part of the GCSE Maths quadratic and simultaneous equations chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page
Chapter summary
| Skill | Method |
|---|---|
| Factorise x2 + bx + c | Find two numbers that multiply to c and add to b: (x + p)(x + q). Take out a common factor first. |
| Difference of two squares | a2 − b2 = (a − b)(a + b) |
| Factorise ax2 + bx + c (Higher) | Multiply a × c, find two numbers with that product that add to b, split the middle term and factorise in pairs. |
| Solve a quadratic by factorising | Rearrange to 0 on one side, factorise, set each bracket to 0 (null factor law), solve, check. |
| Problems leading to quadratics | Choose a letter, form an equation, solve, reject impossible answers, answer in words. |
| Simultaneous equations: elimination | Make one letter match in size, subtract if signs are the same and add if they are different, solve, substitute back, check. |
| Simultaneous equations: substitution | Make a letter the subject of one equation, substitute it (in brackets) into the other, solve, find the other letter, check. |
| Graphical solutions | The solution is where the graphs cross. Roots of a quadratic are where the curve meets y = 0. |
| Equations from words | Choose letters, write one equation per sentence, solve, answer in words with units. |
| Quadratic formula (Higher) | x = (−b ± √(b2 − 4ac)) ÷ 2a; the discriminant decides how many solutions there are. |
| Completing the square (Higher) | x2 + bx + c = (x + b/2)2 − (b/2)2 + c; turning point (−p, q). |
| Linear/quadratic simultaneous (Higher) | Substitute the linear equation into the quadratic, solve, find each second value from the linear equation, write pairs. |
| Quadratics with fractions (Higher) | Multiply every term by the common denominator, rearrange to 0, solve, reject any value that makes a denominator 0. |
Can you…?
- I can factorise quadratics of the form x2 + bx + c and the difference of two squares.
- I can solve quadratic equations by factorising, after rearranging so that one side is 0.
- I can form a quadratic equation from an area or consecutive-number problem and reject impossible solutions.
- I can solve simultaneous linear equations by elimination and by substitution.
- I can solve simultaneous equations and estimate roots of quadratics using graphs.
- I can form and solve simultaneous equations from a word problem.
- Higher: I can factorise ax2 + bx + c, use the quadratic formula, complete the square, solve linear/quadratic simultaneous equations and solve quadratic equations with fractions.
Key terms
- Quadratic expression
- An expression whose highest power is 2, such as x2 + 5x + 6.
- Quadratic equation
- An equation of the form ax2 + bx + c = 0.
- Factorise
- Write an expression as a product of brackets.
- Null factor law
- If a product is 0, at least one of its factors is 0.
- Root
- A solution of an equation. The roots of a quadratic are where its graph crosses the x-axis.
- Difference of two squares
- An expression a2 − b2, which factorises as (a − b)(a + b).
- Simultaneous equations
- Two or more equations that must be true at the same time.
- Elimination
- Adding or subtracting equations to remove one letter.
- Substitution
- Replacing a letter by an expression that equals it.
- Discriminant
- The expression b2 − 4ac; its sign tells how many real solutions a quadratic has.
- Completing the square
- Writing x2 + bx + c as (x + p)2 + q.
- Turning point
- The lowest (minimum) or highest (maximum) point of a quadratic curve.
- Surd
- A square root that is not a whole number, such as √5; an exact answer keeps the surd.