Foundation (Grades 1-5)Higher (Grades 4-9)Calculator AllowedChapter 5-11
What You'll Learn
Master the method of solving quadratic equations by factorizing - one of the most important algebraic skills in GCSE Mathematics. This method uses the null factor law to find two solutions to a quadratic equation.
Why This Matters for GCSE
High-Value Exam Topic: Quadratic equations appear on EVERY GCSE paper (Foundation and Higher). This is a 4-6 mark question that appears multiple times. Common GCSE contexts include:
Finding dimensions of rectangles and shapes (area problems)
Projectile motion and trajectory questions
Number problems ("two consecutive numbers...")
Profit and revenue calculations in business contexts
Graph intersections and coordinate geometry
GCSE Command Words: "Solve", "Find the values of x", "Show that", "Hence or otherwise"
Typical marks: 3-4 marks (Foundation), 4-6 marks (Higher with proof)
The 4-Step Method
1
Rearrange to = 0
Move all terms to one side so the equation equals zero. This is ESSENTIAL for the method to work.
2
Factorize the Quadratic
Write as (x + a)(x + b) = 0 using your factorization skills from Section 4.
3
Use Null Factor Law
If A × B = 0, then A = 0 or B = 0. Set each bracket equal to zero.
4
Solve Each Equation
Solve the two simple linear equations to find both values of x.
Key Concept: The Null Factor Law
If A × B = 0
then A = 0 OR B = 0
This is why we MUST rearrange to equal zero first!
Why does this work?
The only way a product can equal zero is if at least one of the factors is zero. For example:
5 × 0 = 0 ✓
0 × 7 = 0 ✓
But 3 × 4 = 12 (not zero) ✗
Worked Examples
Example 1: Basic Factorization (Foundation - Grade 4/5)
Solve: x² + 5x + 6 = 0
Step 1: Already equals 0 ✓
Step 2: Factorize: Find two numbers that multiply to +6 and add to +5
Numbers are +2 and +3 (because 2 × 3 = 6 and 2 + 3 = 5)
So: (x + 2)(x + 3) = 0
Step 3: Use null factor law:
Either (x + 2) = 0 OR (x + 3) = 0
Step 4: Solve each equation:
x + 2 = 0 → x = -2
x + 3 = 0 → x = -3
Answer: x = -2 or x = -3
Mark Scheme: 1 mark for correct factorization, 1 mark for each solution (3 marks total)
Example 2: Rearranging First (Foundation/Higher - Grade 5)
Solve: x² + 7x = 18
Step 1: Rearrange to equal 0: x² + 7x - 18 = 0
Step 2: Factorize: Find two numbers that multiply to -18 and add to +7
Numbers are +9 and -2 (because 9 × (-2) = -18 and 9 + (-2) = 7)
So: (x + 9)(x - 2) = 0
Step 3: Use null factor law:
Either (x + 9) = 0 OR (x - 2) = 0
Step 4: Solve:
x + 9 = 0 → x = -9
x - 2 = 0 → x = 2
Answer: x = -9 or x = 2
Mark Scheme: 1 mark for rearranging to = 0, 1 mark for factorizing, 1 mark for each solution (4 marks total)
Example 3: Difference of Two Squares (Higher - Grade 6/7)
Solve: x² - 49 = 0
Step 1: Already equals 0 ✓
Step 2: Recognize difference of two squares: a² - b² = (a + b)(a - b)
x² - 49 = x² - 7²
= (x + 7)(x - 7) = 0
Step 3: Use null factor law:
Either (x + 7) = 0 OR (x - 7) = 0
Step 4: Solve:
x + 7 = 0 → x = -7
x - 7 = 0 → x = 7
Answer: x = -7 or x = 7
Mark Scheme: 1 mark for recognizing difference of two squares, 1 mark for each solution (3 marks total)
Example 4: Common Factor First (Higher - Grade 6)
Solve: 2x² + 10x = 0
Step 1: Already equals 0 ✓
Step 2: Take out common factor first: 2x(x + 5) = 0
Step 3: Use null factor law:
Either 2x = 0 OR (x + 5) = 0
Step 4: Solve:
2x = 0 → x = 0
x + 5 = 0 → x = -5
Answer: x = 0 or x = -5
Mark Scheme: 1 mark for taking out common factor, 1 mark for each solution (3 marks total)
Example 5: GCSE Context Question (Higher - Grade 7)
Problem: The area of a rectangle is 48 cm². The length is 2 cm more than the width. Find the width.
Step 2: Factorize: Find numbers that multiply to -48 and add to +2
Numbers are +8 and -6 (because 8 × (-6) = -48 and 8 + (-6) = 2)
(x + 8)(x - 6) = 0
Step 3: x + 8 = 0 OR x - 6 = 0
Step 4: x = -8 or x = 6
Context check: Width cannot be negative, so x = 6 cm
Answer: Width = 6 cm (Length = 8 cm)
Mark Scheme: 1 mark for forming equation, 1 mark for rearranging, 1 mark for factorizing, 1 mark for solutions, 1 mark for rejecting negative answer (5 marks total)
Example 6: When a ≠ 1 (Higher - Grade 8)
Solve: 2x² + 7x + 3 = 0
Step 1: Already equals 0 ✓
Step 2: Factorize (a ≠ 1, so use advanced method)
Find factors of 2 × 3 = 6 that add to 7: these are 6 and 1
Split middle term: 2x² + 6x + 1x + 3 = 0
Group: 2x(x + 3) + 1(x + 3) = 0
= (2x + 1)(x + 3) = 0
Step 3: 2x + 1 = 0 OR x + 3 = 0
Step 4: Solve:
2x + 1 = 0 → 2x = -1 → x = -1/2
x + 3 = 0 → x = -3
Answer: x = -1/2 or x = -3
Mark Scheme: 2 marks for correct factorization (harder when a ≠ 1), 1 mark for each solution (4 marks total)
Practice Questions
Solve these GCSE-style quadratic equations. Give all solutions.
1[3 marks]
Solve: x² + 8x + 15 = 0
2[3 marks]
Solve: x² - 9x + 20 = 0
3[4 marks]
Solve: x² + 3x = 28
4[3 marks]
Solve: x² - 64 = 0
5[3 marks]
Solve: x² - 6x = 0
6[4 marks]
Solve: 3x² - 12x = 0
7[4 marks]
Solve: 2x² + 5x + 2 = 0
8[5 marks]
Problem: A rectangle has an area of 35 cm². Its length is 2 cm more than its width. Find the width of the rectangle.
9[4 marks]
Solve: (x + 3)(x - 5) = 9
Hint: Expand first, then rearrange to = 0
10[5 marks]
Show that the equation x² + 10x + 21 = 0 has solutions x = -3 and x = -7
Hint: Factorize to (x + 3)(x + 7) = 0 and solve to prove the given solutions
Common GCSE Misconceptions
❌ WRONG: x² + 5x + 6 = 2 → (x + 2)(x + 3) = 2 → x = -2 or x = -3 ✅ CORRECT: Must rearrange to = 0 FIRST! x² + 5x + 4 = 0, then factorize
❌ WRONG: (x + 3)(x - 5) = 0 → x = 3 or x = 5 (wrong signs) ✅ CORRECT: x + 3 = 0 → x = -3, and x - 5 = 0 → x = 5
❌ WRONG: x² = 16 → x = 4 (forgetting negative solution) ✅ CORRECT: x² - 16 = 0 → (x + 4)(x - 4) = 0 → x = -4 or x = 4
❌ WRONG: x² + 6x = 0 → x = -6 (dividing by x loses a solution) ✅ CORRECT: x(x + 6) = 0 → x = 0 or x = -6 (two solutions!)
❌ WRONG: Context problem gives x = -5 or x = 6, writing "x = -5 or 6" as final answer ✅ CORRECT: Check context! Length/width cannot be negative, so reject x = -5, answer is 6 cm
GCSE Exam Tips
ALWAYS rearrange to = 0 first! This is the #1 mistake in GCSE exams. The null factor law ONLY works when one side is zero.
Show ALL working: Even if you can factorize in your head, examiners want to see the factorization written out for method marks.
Write both solutions clearly: Use "x = -3 or x = 5" NOT "x = -3, 5" (the comma is ambiguous).
"Solve" vs "Show that": "Solve" means find the answers. "Show that" means prove the given answers are correct (work backwards to check).
Check your solutions: Substitute back into the original equation. If x = 2, does it satisfy x² + 5x - 14 = 0? Quick check!
Context questions: ALWAYS reject impossible solutions (negative lengths, negative time, etc.) and explain why.
Don't forget x = 0: When factorizing x² - 6x = 0, many students forget x = 0 is a solution.
Calculator allowed ≠ calculator required: Factorizing is often faster and more accurate than using the quadratic formula!
Skills Checklist
By the end of this chapter, you should be able to:
✓ Rearrange any quadratic equation to equal zero
✓ Factorize quadratic expressions (revision from Section 4)
✓ Apply the null factor law correctly
✓ Solve simple quadratic equations (Foundation)
✓ Solve quadratics with common factors [Higher]
✓ Solve difference of two squares equations [Higher]
✓ Solve quadratics where a ≠ 1 [Higher]
✓ Form and solve quadratic equations from word problems [Higher]
✓ Reject impossible solutions based on context [Higher]
✓ Prove given solutions using "show that" method [Higher]