Quadratic equations are one of the most important topics in GCSE Maths — they appear in almost every Higher tier paper and are worth a significant number of marks. This guide covers all three methods you need to know: factorising, the quadratic formula, and completing the square.
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the form:
ax² + bx + c = 0
where a ≠ 0. The highest power of x is 2. Examples include:
- x² + 5x + 6 = 0
- 2x² − 3x − 2 = 0
- x² − 4 = 0
- x² + 6x + 9 = 0
A quadratic equation can have two solutions (roots), one repeated solution, or no real solutions — depending on the values of a, b and c.
Method 1 — Solving by Factorising
Factorising is the quickest method when it works. It works cleanly when the solutions are integers or simple fractions.
How it works: You rewrite the quadratic as a product of two brackets, then use the fact that if two things multiply to zero, one of them must be zero.
Worked Example 1
Solve x² + 5x + 6 = 0
Step 1: Find two numbers that multiply to +6 and add to +5. Those numbers are 2 and 3 (2 × 3 = 6, 2 + 3 = 5).
Step 2: Factorise: (x + 2)(x + 3) = 0
Step 3: Set each bracket equal to zero:
- x + 2 = 0 → x = −2
- x + 3 = 0 → x = −3
Answer: x = −2 or x = −3
Worked Example 2
Solve 2x² + 7x − 4 = 0
When the coefficient of x² is not 1, use the AC method.
Step 1: Multiply a × c: 2 × (−4) = −8
Step 2: Find two numbers that multiply to −8 and add to +7. Those numbers are 8 and −1.
Step 3: Split the middle term: 2x² + 8x − x − 4 = 0
Step 4: Group and factorise: 2x(x + 4) − 1(x + 4) = 0 (2x − 1)(x + 4) = 0
Step 5: Solve:
- 2x − 1 = 0 → x = ½
- x + 4 = 0 → x = −4
Answer: x = ½ or x = −4
When to use this method: Try factorising first whenever the numbers look "nice". If you cannot find the factor pair quickly (within about 30 seconds), switch to the quadratic formula.
Method 2 — The Quadratic Formula
The quadratic formula always works, regardless of whether the equation factorises neatly. You must memorise it for the exam — it is not given on the formula sheet.
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
In plain text: x = (−b ± √(b² − 4ac)) ÷ 2a
Worked Example 3
Solve 2x² − 4x − 3 = 0 (give answers to 2 decimal places)
Here a = 2, b = −4, c = −3.
Step 1: Calculate the discriminant: b² − 4ac = (−4)² − 4(2)(−3) = 16 + 24 = 40
Step 2: Apply the formula: x = (4 ± √40) / 4
Step 3: Calculate both values:
- x = (4 + 6.324...) / 4 = 10.324 / 4 = 2.58 (to 2 d.p.)
- x = (4 − 6.324...) / 4 = −2.324 / 4 = −0.58 (to 2 d.p.)
Answer: x = 2.58 or x = −0.58
Important: Always calculate the discriminant (b² − 4ac) first as a separate step. This avoids errors and makes your working easier to follow.
The discriminant tells you:
- b² − 4ac > 0 → two distinct real solutions
- b² − 4ac = 0 → one repeated solution
- b² − 4ac < 0 → no real solutions (the exam will not ask you to solve these at GCSE)
Method 3 — Completing the Square
Completing the square is used to:
- Solve quadratic equations when the formula would be long-winded
- Find the minimum (or maximum) point of a quadratic graph
- Write a quadratic in vertex form
The technique transforms ax² + bx + c into the form a(x + p)² + q.
Worked Example 4
Solve x² + 6x + 7 = 0 by completing the square
Step 1: Halve the coefficient of x: 6 ÷ 2 = 3
Step 2: Write (x + 3)² and expand to check: (x + 3)² = x² + 6x + 9
Step 3: Adjust for the difference: x² + 6x + 7 = (x + 3)² − 9 + 7 = (x + 3)² − 2
Step 4: Set equal to zero and solve: (x + 3)² − 2 = 0 (x + 3)² = 2 x + 3 = ±√2 x = −3 + √2 or x = −3 − √2
In decimal form: x ≈ −1.59 or x ≈ −4.41
When to use this method: Completing the square is especially useful when a question asks you to find the turning point of a quadratic graph, or when b is even (making the halving step clean).
Choosing the Right Method
| Situation | Best method |
|---|---|
| Numbers factorise neatly | Factorising |
| Decimal or surd answers needed | Quadratic formula |
| Question asks for turning point | Completing the square |
| Not sure — use as a backup | Quadratic formula |
Common Mistakes to Avoid
1. Forgetting ± in the formula The ± gives you both solutions. Leaving it out means you only get one answer and will lose marks.
2. Sign errors with b If b is negative, remember that −b is positive. For example, if b = −4, then −b = +4.
3. Not rearranging first The quadratic formula only works when the equation is in the form ax² + bx + c = 0. If you have 3x² = 5x + 2, rearrange to 3x² − 5x − 2 = 0 before applying the formula.
4. Incorrect factorising of 2x² + bx + c Always check by expanding your brackets. If expanding does not give the original equation, your factorisation is wrong.
5. Dividing by x Never divide both sides by x to cancel it. This loses one of the solutions (x = 0). Instead, factorise.
How This Topic Appears in the Exam
At Higher tier, quadratic equations typically appear in the following ways:
- A direct "solve this quadratic equation" question (2–3 marks)
- A problem where you form and solve a quadratic from a word problem or geometry context
- A question asking you to find where two graphs intersect (requires solving a simultaneous equation where one is quadratic)
- A question asking you to find the turning point of a quadratic graph (completing the square)
At Foundation tier, simple factorising of x² + bx + c = 0 (where a = 1) may appear.
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