Short answer
At GCSE you have four methods for solving a quadratic equation: factorising, the quadratic formula, completing the square, and reading solutions from a graph. The method you choose depends on how tidy the coefficients are and what the question asks for.
At a glance
- Key stage
- GCSE
- Subject
- Algebra
- Type
- Guide
- For
- Students
- Read time
- 5 min
- Last updated
- 21 October 2026
Where this fits
- Key Stage 3Years 7–9
- GCSEYears 10–11This article
What form must a quadratic be in before you solve it?
Always rearrange to the form ax² + bx + c = 0 before choosing a method. A quadratic equal to zero is the standard starting point for all methods.
Example: 3x² + 5 = 2x + 9 → rearrange → 3x² − 2x − 4 = 0 ✓
Once in this form, identify a, b, and c, and then decide your method.
Method 1: Factorising
Use when: the quadratic factorises neatly with integer roots (small, whole-number values that multiply and add correctly).
Quick check: look for two numbers that multiply to ac and add to b. If you find them quickly, factorising is the fastest route.
Best suited to: a = 1 or small integer a, with small values of b and c.
Example: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3. ✓ Fast.
Avoid when: the discriminant (b² − 4ac) is not a perfect square, or when the coefficients are large — factorising will take longer than the formula.
Method 2: The quadratic formula
Use when: the quadratic does not factorise easily, when the question asks for answers to a given number of decimal places, or when coefficients are messy.
Formula: x = (−b ± √(b² − 4ac)) ÷ (2a)
This always works for any quadratic with real solutions — it is the failsafe method.
Example: 2x² + 3x − 4 = 0 (a = 2, b = 3, c = −4).
Discriminant = 9 + 32 = 41 (not a perfect square — formula is the right choice).
x = (−3 ± √41) ÷ 4 → x ≈ 0.85 or x ≈ −2.35 (2 d.p.)
When to use it by default: if the question says "give your answer to 2 decimal places" or "give your answer in the form p ± q√r," the formula is almost always intended.
Method 3: Completing the square
Use when: the question says "write in the form (x + p)² + q" or asks for the vertex (minimum or maximum point) of the parabola.
Completing the square also solves the equation, but is more powerful for finding the turning point.
Example: x² + 6x + 7 = 0.
Complete the square: (x + 3)² − 9 + 7 = 0 → (x + 3)² = 2 → x + 3 = ±√2 → x = −3 ± √2.
Vertex of y = x² + 6x + 7 is at (−3, −2). This is the key advantage over the formula.
Avoid when: a ≠ 1 (completing the square with a ≠ 1 is trickier and the formula is quicker).
Method 4: Graphical method
Use when: asked to "solve graphically" or when an approximate answer from a graph is acceptable.
Draw y = ax² + bx + c (or the rearranged form) on a coordinate grid and read off the x-values where the curve crosses the x-axis.
Accuracy: graphical solutions are approximate — typically to 1 decimal place. They are never as precise as algebraic methods.
Use also to: check the number of solutions (two, one, or no intersections with the x-axis).
Summary: which method to use?
| Situation | Best method |
|---|---|
| Coefficients are small integers | Try factorising first |
| Asked for "exact form" (e.g. 2 ± √3) | Completing the square or formula |
| Asked to 2 d.p. or 3 s.f. | Quadratic formula |
| Factorising fails or takes too long | Quadratic formula |
| Asked for the vertex / turning point | Completing the square |
| Asked to "solve graphically" | Graphical method |
| a ≠ 1 | Usually formula (or careful completing the square) |
What if you are not sure whether a quadratic factorises?
Check the discriminant b² − 4ac:
- If b² − 4ac is a perfect square (0, 1, 4, 9, 16, …) the quadratic factorises over the integers.
- If not, use the formula.
Example: x² + 5x + 3 = 0. Discriminant = 25 − 12 = 13. Not a perfect square → use the formula.
Example: x² − 7x + 12 = 0. Discriminant = 49 − 48 = 1. Perfect square → factorises: (x − 3)(x − 4) = 0. ✓
Frequently asked questions
Can I always use the formula, even if the quadratic factorises?
Yes — the formula always gives correct answers. Factorising is quicker when it works, but using the formula is never wrong. In an exam, if you cannot see how to factorise within 30 seconds, switch to the formula.
Why does completing the square give the vertex but the formula does not?
The quadratic formula is derived from completing the square — it skips straight to the x-intercepts. When you complete the square in the form (x + p)² + q = 0, the point (−p, q) is the vertex. The formula discards this intermediate step, so you lose the vertex information.
Is the graphical method accepted in GCSE mark schemes?
Only when the question specifically says "solve graphically" or "by drawing a graph." Algebraic methods will always earn full marks; a graphical answer when an exact answer is expected will not.
Does it matter which method I use if the question just says "solve"?
No — any valid method that reaches the correct solutions earns full marks. Choose the method you are most confident with, but bear in mind that factorising and the formula are the most reliable and quickest under exam conditions.
Professor Pi teaches you when and how to use every quadratic method — add the AI Tutors connector at aitutors.me.
Key terms
- Use when
- Quick check
- ac
- Best suited to
- Avoid when
- vertex
- Accuracy
- Use also to