Algebra · GCSE Maths
Quadratic equations
Solve ax² + bx + c = 0 by factorising, completing the square, or the quadratic formula, and know how many roots to expect.
A quadratic is a U or an n. Factorise if you can, use the formula if you cannot, and complete the square when the turning point matters.
The important bits
What you need to know
- 1
A quadratic equation can be written as ax² + bx + c = 0 with a ≠ 0. Rearrange to this form before you choose a method.
- 2
If it factorises, write (px + q)(rx + s) = 0, then each bracket can be zero: two linear equations, two roots (which may be repeated).
- 3
The quadratic formula is x = [−b ± √(b² − 4ac)] / (2a). Substitute carefully, keeping the ± until the last step.
- 4
The discriminant D = b² − 4ac tells you the number of real roots: D > 0 two, D = 0 one repeated, D < 0 none on the GCSE course.
- 5
Completing the square writes x² + bx as (x + b/2)² − (b/2)². It gives the turning point and a route to the roots.
- 6
The graph of y = ax² + bx + c is a parabola. If a > 0 it is a U; if a < 0 it is an n. Roots are x-intercepts.
- 7
A quadratic and a linear graph meet where you solve the quadratic formed by setting them equal. The algebra is the intersection.
- 8
Check each root in the original equation. Extraneous roots appear if you squared both sides earlier in a related problem.
Go deeper
Factorising is a product that equals zero
The zero-product property is the reason factorising works: if two numbers multiply to zero, at least one of them is zero. That is not true of 12. So x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0, hence x = −2 or x = −3. You must reach = 0 first. If the question gives x² + 5x = −6, add 6 before you factorise. For 2x² + 7x + 3, look for two numbers that multiply to 2 × 3 = 6 and add to 7, namely 6 and 1, then split the middle term or use grouping: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Expand as a check every time. If a does not factorise nicely, stop hunting and reach for the formula; five minutes of false pairs wastes the paper.
Go deeper
The formula is substitution with discipline
Write a, b and c from ax² + bx + c = 0, including signs. For 2x² − 3x − 5 = 0, a = 2, b = −3, c = −5. Then b² − 4ac = 9 + 40 = 49. The square root is 7. So x = [3 ± 7] / 4, which gives x = 2.5 or x = −1. The minus in −b is the first trap: −(−3) is +3. The second trap is dividing only the square root by 2a instead of the whole numerator. Write the formula once, substitute in brackets, then simplify. If the discriminant is not a square, leave the answer in surd form unless the question asks for decimals. Rounding to two decimal places too early is how two otherwise correct roots miss the final mark.
Go deeper
Completing the square shows the graph
x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4. The turning point is (−3, −4), because a square is never negative and is zero when x = −3. Setting (x + 3)² − 4 = 0 gives (x + 3)² = 4, so x + 3 = ±2, hence x = −1 or x = −5. Completing the square is therefore both a solving method and a graph method. For ax² + bx + c with a ≠ 1, factor out a first: 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5. The same picture, scaled. Higher-tier papers use this form to ask for the minimum value or the line of symmetry without plotting a table of values.
See the idea in action
Solve x² − 5x + 6 = 0. Factorise: (x − 2)(x − 3) = 0, so x = 2 or x = 3. Check: 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0. The same roots come from the formula with a = 1, b = −5, c = 6, because the discriminant is 25 − 24 = 1.
Exam technique
Turn knowledge into marks
If the quadratic looks friendly, factorise. If the numbers are ugly, write a, b and c, then the formula. Do not mix methods halfway and drop a sign.
Common mistakes
Do not give these marks away
- 01
Forgetting to rearrange to = 0 before factorising, so the brackets do not correspond to the roots.
- 02
Using b as positive when it is negative in the formula, especially −b when b is already negative.
- 03
Writing only one solution when a factorised quadratic has two distinct linear factors.
The solutions of x² − 5x + 6 = 0 are
Ax = −2 and x = −3
Bx = 2 and x = 3
Cx = 1 and x = 6
Dx = −1 and x = 6
Show the answer
x = 2 and x = 3. (x − 2)(x − 3) = 0, so x = 2 or x = 3. The signs in the brackets are the opposite of the roots.
Quick questions
If this is the bit you searched
When must I use the quadratic formula?
When the quadratic does not factorise over the integers, or when the question asks for answers to a given number of decimal places. Factorising is quicker when it works.
What does a negative discriminant mean at GCSE?
No real roots. The graph does not meet the x-axis. You are not required to give complex solutions on the GCSE course.