Solving quadratic equations

A quadratic equation has the form ax² + bx + c = 0. Edexcel 1MA1 tests all three solution methods, and Higher papers also require completing the square for exact solutions.

Method 1 — Factorising

Works when the quadratic factors neatly.

Step: find two numbers that multiply to give ac and add to give b. Split the middle term, group, and factorise.

Example: x² + 5x + 6 = 0. Numbers that multiply to 6 and add to 5: 2 and 3. (x + 2)(x + 3) = 0 → x = −2 or x = −3.

Example (leading coefficient ≠ 1): 2x² + 7x + 3 = 0. ac = 6; pairs: 1 × 6 = 6, sum = 7 ✓. 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1) = 0. x = −3 or x = −½.

Method 2 — Quadratic formula

Always works. Edexcel expects it when factorising fails.

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

The discriminant b² − 4ac determines the number of real solutions:

• b² − 4ac > 0: two distinct real solutions.
• b² − 4ac = 0: one repeated solution.
• b² − 4ac < 0: no real solutions.

Example: 2x² − 5x − 3 = 0. a=2, b=−5, c=−3. Discriminant = 25 + 24 = 49. x = (5 ± 7)/4 → x = 3 or x = −½.

Method 3 — Completing the square

Converts to (x + p)² + q = 0 form.

For x² + bx + c = 0: Step 1: (x + b/2)² − (b/2)² + c = 0. Step 2: solve for x.

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

Edexcel often asks: "Give your answer in the form p ± q√r" — completing the square is the intended route.

Forming quadratics from context

Edexcel 1MA1 Paper 2/3 frequently frames quadratics as area problems, consecutive-integer problems, or ratio problems. Read carefully, set up the equation, solve, then check whether both roots are valid in context (e.g. length must be positive).