M15 · Quadratic equations | Edexcel GCSE Maths

Revise the key ideas

A quadratic equation contains a squared unknown as its highest power. Write it as ax² + bx + c = 0, where a ≠ 0, before choosing a solution method.
If two factors multiply to zero, at least one factor must be zero. This zero-product rule applies to a product, not a sum.
Worked example: x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0. Therefore x = −2 or x = −3. Check each in the original equation.
Worked example: x² = 5x gives x² − 5x = x(x − 5) = 0, so x = 0 or x = 5. Dividing straight away by x would lose the zero solution.
For x² = 49, x = ±7. Taking a square root without considering both signs can lose a solution.
Use a graph of y = ax² + bx + c to find approximate roots at the x-axis. A parabola may have two, one repeated, or no real roots.
Roots of x squared plus five x plus six
Enlarge diagram: Roots of x squared plus five x plus six

Worked example: A rectangle has width x cm and length x + 3 cm, and area 28 cm². x(x + 3) = 28 gives (x + 7)(x − 4) = 0. Only x = 4 cm is a valid width; a negative length is rejected with a reason.
Do not discard a negative algebraic solution unless the context rules it out. Coordinates, temperatures and some other quantities can be negative.
When using an approximate root, retain enough precision for subsequent work and round the final contextual answer appropriately.

For x² + bx, add and subtract (b2)² to create a square. x² + 6x + 5 = (x + 3)² − 4.
Worked example: x² + 6x + 5 = 0 gives (x + 3)² = 4, so x + 3 = ±2 and x = −1 or −5.

For ax² + bx + c = 0, x = −b ± √(b² − 4ac)2a. The denominator divides the whole numerator, including the square-root term.
Worked example: 2x² − 3x − 1 = 0 has a = 2, b = −3, c = −1. x = 3 ± √174 ≈ 1.781 or −0.281. Use brackets around negative substitutions.
The discriminant b² − 4ac is positive for two distinct real roots, zero for a repeated root and negative for no real roots. GCSE real-number work cannot take the square root of a negative discriminant.
Choose an efficient method: factorise where possible, use completing the square for exact structure, or use the formula for a general quadratic. A calculator result still needs clear working when asked.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.