Solving Quadratic Equations

A quadratic equation has the form $ax^2 + bx + c = 0$ where $a \neq 0$.

There are three main methods for solving quadratics: factorising, completing the square, and the quadratic formula. WJEC Eduqas papers require you to know all three.

Method 1: Factorising

Works when the quadratic factorises neatly. Best for integer solutions.

Steps:

1. Rearrange to $ax^2 + bx + c = 0$.
2. Find two numbers that multiply to $ac$ and add to $b$.
3. Write as a product of two brackets and solve.

Example: Solve $x^2 + 5x + 6 = 0$

Find two numbers multiplying to 6 and adding to 5: these are 2 and 3. $$(x + 2)(x + 3) = 0$$ $$x = -2 \quad \text{or} \quad x = -3$$

Example with a ≠ 1: Solve $2x^2 + 7x + 3 = 0$

$ac = 6$; find numbers multiplying to 6 adding to 7: 1 and 6. $$2x^2 + x + 6x + 3 = 0 \implies x(2x + 1) + 3(2x + 1) = 0$$ $$(x + 3)(2x + 1) = 0$$ $$x = -3 \quad \text{or} \quad x = -\tfrac{1}{2}$$

Method 2: Completing the Square

Used when factorising is not possible or when the question asks for the minimum/maximum of a quadratic function.

For $x^2 + bx + c$: $$x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$$

Example: Solve $x^2 + 6x + 2 = 0$

Complete the square: $(x + 3)^2 - 9 + 2 = 0$ $$(x + 3)^2 = 7$$ $$x + 3 = \pm\sqrt{7}$$ $$x = -3 \pm \sqrt{7}$$

Answers: $x \approx -0.354$ or $x \approx -5.646$

Method 3: The Quadratic Formula

Use when the question says "give your answer to 2 decimal places" or when factorising fails.

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

Example: Solve $3x^2 - 5x - 2 = 0$

$a = 3$, $b = -5$, $c = -2$

$$x = \frac{5 \pm \sqrt{25 + 24}}{6} = \frac{5 \pm \sqrt{49}}{6} = \frac{5 \pm 7}{6}$$ $$x = 2 \quad \text{or} \quad x = -\frac{1}{3}$$

The Discriminant

The expression $b^2 - 4ac$ under the square root is the discriminant, $\Delta$:

$\Delta$	Nature of roots
$\Delta > 0$	Two distinct real roots
$\Delta = 0$	One repeated root
$\Delta < 0$	No real roots

WJEC Exam Tips

• Always rearrange to $ax^2 + bx + c = 0$ before applying any method.
• If the question asks for an exact answer, leave as $x = a \pm \sqrt{b}$.
• If asked to give to 2 d.p., use the formula and a calculator.
• Check solutions by substituting back.