Solving quadratic equations

Three methods for solving ax² + bx + c = 0 are assessed on OCR J560. Higher-tier papers often specify which method to use — know all three.

Method 1: Factorising

1. Rearrange to ax² + bx + c = 0.
2. Factorise the left side.
3. Set each factor to zero.

Example: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3.

Example: x² − 9 = 0 → (x+3)(x−3) = 0 → x = ±3.

When to use: when factorisation is straightforward (small integer roots).

Method 2: The quadratic formula

For ax² + bx + c = 0:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

x = (−b ± √(b² − 4ac)) / (2a)

The discriminant is b² − 4ac:

• > 0: two distinct real roots.
• = 0: one repeated root (the vertex touches the x-axis).
• < 0: no real roots (the curve doesn't cross the x-axis).

Example: 2x² + 5x − 3 = 0. a=2, b=5, c=−3. x = (−5 ± √(25+24)) / 4 = (−5 ± √49) / 4 = (−5 ± 7) / 4. x = 2/4 = 0.5 or x = −12/4 = −3.

Method 3: Completing the square

Rewrite x² + bx + c = 0 as (x + b/2)² − (b/2)² + c = 0.

Example: x² + 6x + 2 = 0. → (x + 3)² − 9 + 2 = 0 → (x + 3)² = 7 → x + 3 = ±√7 → x = −3 ± √7.

For ax² + bx + c = 0 where a ≠ 1: divide through by a first.

When to use completing the square: when the question asks for it; when writing in vertex form for a graph question; when exact surd answers are needed.

Choosing a method

Situation	Method
Obvious factors	Factorising
Question specifies formula	Formula
Need exact surd answer	Completing the square or formula
Non-integer coefficients	Formula

Common OCR exam mistakes

1. Not rearranging to = 0 before factorising or using the formula.
2. Forgetting ±√ — quadratics usually have TWO solutions.
3. Arithmetic errors in the discriminant — calculate b² and 4ac separately.
4. When completing the square with a ≠ 1: not dividing through by a first.