Higher Algebra – Quadratics & Formula

Quadratic formula, completing the square, and solving equations with algebraic fractions – essential Higher skills.

Higher tier requires the quadratic formula and often completing the square. These methods work when factorisation is difficult or impossible.

Practice lesson for the WJEC Mathematics and Numeracy Double Award. This is not an official WJEC page.

Double Award guide·WJEC / CBAC

The quadratic formula

For ax² + bx + c = 0: x = [-b ± √(b² − 4ac)] / (2a) Discriminant b² − 4ac: • > 0 → two real solutions • = 0 → one real solution (repeated root) • < 0 → no real solutions

Worked example

Solve x² − 6x + 5 = 0 using the formula.

Solution

a = 1, b = −6, c = 5 x = [6 ± √(36 − 20)] / 2 = [6 ± √16] / 2 = [6 ± 4] / 2 **x = 5** or **x = 1**

Completing the square

For x² + bx + c: x² + bx = (x + b/2)² − (b/2)² Then rearrange to solve or to find turning point.

Worked example

Write x² + 8x + 10 in completed square form.

Solution

x² + 8x + 10 = (x + 4)² − 16 + 10 = **(x + 4)² − 6**

Practice questions

Try these without looking at the answers first.

Q1.Solve using the formula: x² + 5x + 6 = 0

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x = −2 or x = −3

Q2.Complete the square: x² − 6x + 2

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(x − 3)² − 7

Q3.Discriminant of x² − 4x + 7? How many real roots?

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−12, no real roots

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