How to Solve Quadratic Equations (Factorising, Quadratic Formula, Completing the Square)

How to Solve Quadratic Equations (Factorising, Quadratic Formula, Completing the Square)

GCSE / A-Level Maths – Exam-Ready

Introduction

"Mastering quadratics unlocks 10–15% of your GCSE Maths exam marks—and real-world problems like calculating projectile motion, profit maximisation, or even designing bridges. One question could be the difference between a 6 and a 7, or a B and an A. Let’s make sure you never lose marks here again."

What You Need To Know First

1. Expanding brackets – You must be able to multiply out expressions like (x + 3)(x – 2).
2. Solving linear equations – If you can solve 2x + 5 = 0, you’re ready for quadratics.
3. Square roots and negative numbers – Know that √9 = ±3 and –3 × –3 = 9.

Key Vocabulary

Formulas To Know

1. Quadratic Formula
   x = [–b ± √(b² – 4ac)] / (2a)
2. a = coefficient of x²
3. b = coefficient of x
4. c = constant term

5. MEMORISE THIS – It’s your safety net when factorising fails.

6. Discriminant
   D = b² – 4ac

7. If D > 0: Two distinct real roots.
8. If D = 0: One real root (repeated).
9. If D < 0: No real roots (complex numbers at A-Level).

10. Given on exam sheet (but know how to use it).

11. Completing the Square Form
    x² + bx = (x + b/2)² – (b/2)²

12. MEMORISE THIS – It’s the foundation for the quadratic formula.

Step-by-Step Method

Method 1: Factorising (Fastest – Use First)

When to use: When the quadratic can be split into two simple brackets (e.g., x² + 5x + 6 = 0).

1. Write the equation in standard form: ax² + bx + c = 0.
2. Find two numbers that:
3. Multiply to a × c (the first and last coefficients).
4. Add to b (the middle coefficient).
5. Split the middle term using these two numbers.
6. Factor by grouping – Take out common factors from the first two and last two terms.
7. Set each bracket to zero and solve for x.

Worked Example: Solve x² + 5x + 6 = 0. 1. Already in standard form. 2. Numbers that multiply to 6 and add to 5: 2 and 3. 3. Split: x² + 2x + 3x + 6 = 0. 4. Group: (x² + 2x) + (3x + 6) = x(x + 2) + 3(x + 2) = (x + 2)(x + 3) = 0. 5. Solutions: x + 2 = 0 → x = –2 or x + 3 = 0 → x = –3.

What we did and why: We rewrote the quadratic as two brackets multiplied together, then used the zero product property (if A × B = 0, then A = 0 or B = 0) to find the roots.

Method 2: Quadratic Formula (Foolproof – Use When Factorising Fails)

When to use: When the quadratic doesn’t factorise easily (e.g., 2x² + 3x – 7 = 0).

1. Write the equation in standard form: ax² + bx + c = 0.
2. Identify a, b, and c.
3. Calculate the discriminant: D = b² – 4ac.
4. If D < 0, stop here (no real roots at GCSE).
5. Plug into the quadratic formula:
   x = [–b ± √D] / (2a).
6. Simplify the square root (if possible).
7. Write two solutions (one with +, one with –).

Worked Example: Solve 2x² + 3x – 7 = 0. 1. Standard form: 2x² + 3x – 7 = 0. 2. a = 2, b = 3, c = –7. 3. Discriminant: D = 3² – 4(2)(–7) = 9 + 56 = 65 (positive, so two real roots). 4. Formula: x = [–3 ± √65] / 4. 5. Solutions: x = (–3 + √65)/4 or x = (–3 – √65)/4.

What we did and why: We used the quadratic formula as a universal method when factorising was too hard. The discriminant told us there were two real roots before we even solved.

Method 3: Completing the Square (Useful for Graphs & A-Level)

When to use: When asked to write in vertex form (y = a(x – h)² + k) or for A-Level questions on transformations.

1. Write the equation in standard form: x² + bx + c = 0 (if a ≠ 1, divide everything by a first).
2. Move c to the other side: x² + bx = –c.
3. Complete the square:
4. Take half of b, square it, and add to both sides.
5. Rewrite the left side as (x + b/2)².
6. Solve for x by taking square roots and isolating x.

Worked Example: Solve x² + 6x + 2 = 0 by completing the square. 1. Standard form: x² + 6x + 2 = 0. 2. Move c: x² + 6x = –2. 3. Complete the square:
- Half of 6 is 3, squared is 9.
- Add 9 to both sides: x² + 6x + 9 = 7.
- Rewrite: (x + 3)² = 7. 4. Solve: x + 3 = ±√7 → x = –3 ± √7.

What we did and why: We rewrote the quadratic in perfect square form to make it easy to solve. This method is also key for sketching parabolas (A-Level).

Worked Examples

Example 1 – Basic (Factorising)

Question: Solve x² – 5x + 6 = 0. Solution: 1. Standard form: x² – 5x + 6 = 0. 2. Numbers that multiply to 6 and add to –5: –2 and –3. 3. Factorise: (x – 2)(x – 3) = 0. 4. Solutions: x = 2 or x = 3.

What we did and why: We used factorising because the numbers were simple. Always check if the quadratic can be split into two brackets first—it’s the fastest method.

Example 2 – Medium (Quadratic Formula)

Question: Solve 3x² – 2x – 4 = 0. Give answers to 2 decimal places. Solution: 1. Standard form: 3x² – 2x – 4 = 0. 2. a = 3, b = –2, c = –4. 3. Discriminant: D = (–2)² – 4(3)(–4) = 4 + 48 = 52. 4. Formula: x = [2 ± √52] / 6. 5. Simplify: √52 = 2√13 → x = [2 ± 2√13] / 6 = [1 ± √13] / 3. 6. Decimal answers: x ≈ 1.54 or x ≈ –0.87.

What we did and why: We used the quadratic formula because factorising wasn’t obvious. The discriminant told us there were two real roots, and we simplified the surd before calculating decimals.

Example 3 – Exam-Style (Completing the Square + Context)

Question (A-Level): A ball is thrown upwards. Its height h metres after t seconds is given by h = –5t² + 20t + 1. Find the time when the ball hits the ground (h = 0). Give your answer in surd form. Solution: 1. Set h = 0: –5t² + 20t + 1 = 0. 2. Divide by –5: t² – 4t – 0.2 = 0. 3. Complete the square:
- Move c: t² – 4t = 0.2.
- Half of –4 is –2, squared is 4.
- Add 4: t² – 4t + 4 = 4.2 → (t – 2)² = 4.2. 4. Solve: t – 2 = ±√4.2 → t = 2 ± √4.2. 5. Reject negative time: t = 2 + √4.2 seconds.

What we did and why: We completed the square because the question asked for an exact answer (surd form). The context (ball hitting the ground) meant we discarded the negative solution.

Common Mistakes

Exam Traps

1-Minute Recap

"You’ve got three tools for quadratics: factorising, the quadratic formula, and completing the square. Here’s the game plan: 1. Always try factorising first—it’s the fastest. Look for two numbers that multiply to a × c and add to b. 2. If factorising fails, use the quadratic formula. Memorise it: x = [–b ± √(b² – 4ac)] / (2a). Calculate the discriminant first to check for real roots. 3. Completing the square is for A-Level or when you need the vertex form. Half b, square it, and add it to both sides. 4. Watch for traps: Rearrange to standard form, check for rounding instructions, and don’t forget the ±! 5. Practice all three methods—exam questions will test which one you pick. Now go smash those quadratics!