GCSE Maths Algebra How to Solve Quadratic Equations (Factorising, Quadratic Formula, Completing the Square) – Complete Guide

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

Hook: "Mastering quadratics unlocks projectile motion in Physics, reaction rates in Chemistry, and population models in Biology—plus 12-15 marks in your GCSE/A-Level exams. This guide gives you the exact steps to solve any quadratic equation, fast and error-free."

What You Need to Know First

1. Expanding brackets (e.g., (x + 3)(x – 2) = x² + x – 6)
2. Solving linear equations (e.g., 2x + 5 = 0 → x = –2.5)
3. Basic algebra rules (e.g., moving terms across the equals sign changes the sign)

Key Terms & Formulas

1. Standard Form of a Quadratic Equation

Formula: ax² + bx + c = 0 - a, b, c = constants (a ≠ 0) - x = variable (unknown)

2. Factorising (Factoring)

Method: Rewrite ax² + bx + c as (px + q)(rx + s) = 0 - MEMORISE THIS: If (px + q)(rx + s) = 0, then x = –q/p or x = –s/r

3. Quadratic Formula

Formula: x = [–b ± √(b² – 4ac)] / (2a) - MEMORISE THIS (or check if given on your exam sheet) - b² – 4ac = discriminant (tells you how many solutions exist)

4. Completing the Square

Formula: ax² + bx + c = a(x + d)² + e - d = b/(2a) - e = c – (b²)/(4a) - MEMORISE THE STEPS (not the final formula)

Step-by-Step Method

Method 1: Factorising (Best for Simple Equations)

When to use: a = 1 or small integers (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
4. Add to b
5. Rewrite the middle term using these numbers:
6. x² + (sum)x + (product) = 0
7. Factor by grouping:
8. (x + first number)(x + second number) = 0
9. Set each bracket to zero and solve for x:
10. x + first number = 0 → x = –first number
11. x + second number = 0 → x = –second number

Worked Example (Factorising): Solve x² + 7x + 12 = 0 1. Standard form: x² + 7x + 12 = 0 (already correct) 2. Find two numbers that multiply to 12 and add to 7 → 3 and 4 3. Rewrite: x² + 3x + 4x + 12 = 0 4. Factor: (x + 3)(x + 4) = 0 5. Solve:
- x + 3 = 0 → x = –3
- x + 4 = 0 → x = –4 Answer: x = –3 or x = –4

What we did and why: We split the middle term (7x) into 3x + 4x because 3 × 4 = 12 (the constant term) and 3 + 4 = 7 (the coefficient of x). This lets us factor easily.

Method 2: Quadratic Formula (Works for Any Equation)

When to use: a ≠ 1 or factorising is too hard (e.g., 2x² – 5x + 1 = 0)

1. Write the equation in standard form: ax² + bx + c = 0
2. Identify a, b, c (include signs!)
3. Calculate the discriminant: D = b² – 4ac
4. If D > 0 → 2 real solutions
5. If D = 0 → 1 real solution
6. If D < 0 → no real solutions (complex numbers)
7. Plug into the quadratic formula:
   x = [–b ± √D] / (2a)
8. Simplify the square root and fraction.

Worked Example (Quadratic Formula): Solve 2x² – 4x – 3 = 0 1. Standard form: 2x² – 4x – 3 = 0 2. a = 2, b = –4, c = –3 3. Discriminant: D = (–4)² – 4(2)(–3) = 16 + 24 = 40 4. Plug into formula:
x = [–(–4) ± √40] / (2 × 2) = [4 ± √40] / 4 5. Simplify:
- √40 = 2√10
- x = [4 ± 2√10] / 4 = (2 ± √10)/2 Answer: x = (2 + √10)/2 or x = (2 – √10)/2

What we did and why: We used the quadratic formula because factorising 2x² – 4x – 3 is tricky. The discriminant (40) told us there are two real solutions. We simplified √40 to 2√10 to make the answer cleaner.

Method 3: Completing the Square (Useful for Vertex Form)

When to use: When asked for the vertex form (e.g., y = a(x + d)² + e) or if a = 1

1. Write the equation in standard form: ax² + bx + c = 0
2. Divide all terms by a (if a ≠ 1):
   x² + (b/a)x + (c/a) = 0
3. Move the constant term to the other side:
   x² + (b/a)x = –(c/a)
4. Complete the square:
5. Take half of b/a, square it, and add to both sides.
6. (x + (b/2a))² = (b/2a)² – (c/a)
7. Take the square root of both sides:
   x + (b/2a) = ±√[(b/2a)² – (c/a)]
8. Solve for x.

Worked Example (Completing the Square): Solve x² + 6x + 2 = 0 1. Standard form: x² + 6x + 2 = 0 (a = 1) 2. Move constant: x² + 6x = –2 3. Complete the square:
- Half of 6 is 3, square it → 9
- Add 9 to both sides: x² + 6x + 9 = –2 + 9
- (x + 3)² = 7 4. Square root: x + 3 = ±√7 5. Solve: x = –3 ± √7 Answer: x = –3 + √7 or x = –3 – √7

What we did and why: We completed the square to rewrite x² + 6x + 2 in vertex form. This method is useful for finding the minimum/maximum of a quadratic (e.g., in projectile motion).

Worked Examples

Example 1 – Basic (Factorising)

Solve x² – 5x + 6 = 0 1. Standard form: x² – 5x + 6 = 0 2. Find two numbers that multiply to 6 and add to –5 → –2 and –3 3. Factor: (x – 2)(x – 3) = 0 4. Solve:
- x – 2 = 0 → x = 2
- x – 3 = 0 → x = 3 Answer: x = 2 or x = 3

What we did and why: We looked for two numbers that multiply to 6 and add to –5. Factoring is the fastest method here because a = 1.

Example 2 – Medium (Quadratic Formula)

Solve 3x² + 2x – 1 = 0 1. Standard form: 3x² + 2x – 1 = 0 2. a = 3, b = 2, c = –1 3. Discriminant: D = 2² – 4(3)(–1) = 4 + 12 = 16 4. Plug into formula:
x = [–2 ± √16] / 6 = [–2 ± 4] / 6 5. Solutions:
- x = (–2 + 4)/6 = 2/6 = 1/3
- x = (–2 – 4)/6 = –6/6 = –1 Answer: x = 1/3 or x = –1

What we did and why: We used the quadratic formula because factorising 3x² + 2x – 1 is not straightforward. The discriminant (16) told us there are two real solutions.

Example 3 – Exam-Style (Completing the Square)

The height h (in metres) of a ball after t seconds is given by h = –5t² + 20t + 1. When does the ball hit the ground? 1. Set h = 0: –5t² + 20t + 1 = 0 2. Divide by –5: t² – 4t – 0.2 = 0 3. Move constant: t² – 4t = 0.2 4. Complete the square:
- Half of –4 is –2, square it → 4
- Add 4 to both sides: t² – 4t + 4 = 4.2
- (t – 2)² = 4.2 5. Square root: t – 2 = ±√4.2 6. Solve: t = 2 ± √4.2 7. Reject negative time: t = 2 + √4.2 ≈ 4.05 s Answer: The ball hits the ground at t ≈ 4.05 seconds.

What we did and why: We completed the square to solve for t because the equation was not easily factorable. We rejected the negative solution because time cannot be negative.

Common Mistakes

Exam Traps

1-Minute Recap (Night Before the Exam)

"Listen up—quadratics are 12-15 marks in your exam, so nail this now. Here’s the game plan: 1. Always write the equation in standard form first: ax² + bx + c = 0. 2. Try factorising first—if a = 1 and the numbers are small, it’s the fastest way. 3. If factorising fails, use the quadratic formula. Memorise it: x = [–b ± √(b² – 4ac)] / (2a). Plug in a, b, c carefully—watch those signs! 4. Completing the square? Only if the question asks for it. Divide by a, move the constant, then add (b/2)² to both sides. 5. Check your discriminant—if it’s negative, there are no real solutions. 6. Simplify your answers—leave surds as √5, not 2.236. 7. Reject impossible solutions—time can’t be negative, lengths can’t be negative.

You’ve got this. Now go solve some quadratics!"