What are quadratic expressions and equations?
A quadratic expression is any expression of the form ax² + bx + c, where a ≠ 0. You will need to expand, simplify and factorise these expressions throughout your GCSE course.
A quadratic equation sets a quadratic expression equal to zero: ax² + bx + c = 0. Solving these equations — finding the values of x — is one of the most tested skills at GCSE. There are four methods you need to know:
- Factorising (the fastest method when it works)
- Completing the square (used to find the vertex of a parabola)
- The quadratic formula (works for any quadratic)
- Graphically (reading roots from a sketch)
This page covers each method with worked examples and graded practice questions.
Expanding double brackets
Before you can factorise a quadratic, you need to be confident expanding two brackets. Use FOIL: First, Outer, Inner, Last. Every term in the first bracket multiplies every term in the second.
Worked Example 1 — Basic double brackets
Expand and simplify: (x + 5)(x − 3)
5
Collect like terms: x² − 3x + 5x − 15 = x² + 2x − 15
Answerx² + 2x − 15
Worked Example 2 — Perfect square
Expand and simplify: (3x − 2)²
1
Write as two brackets: (3x − 2)(3x − 2)
2
FOIL: 9x² − 6x − 6x + 4 = 9x² − 12x + 4
Answer9x² − 12x + 4
Factorising quadratics
Factorising is the reverse of expanding. For a quadratic ax² + bx + c, find two numbers that multiply to ac and add to b. When a = 1 this simplifies greatly.
Worked Example 3 — Factorising when a = 1
Factorise: x² − x − 12
1
Find two numbers that multiply to −12 and add to −1: −4 and +3
2
Write in factorised form: (x − 4)(x + 3)
3
Check by expanding: x² + 3x − 4x − 12 = x² − x − 12 ✓
Answer(x − 4)(x + 3)
Worked Example 4 — Factorising when a ≠ 1
Factorise: 3x² + 11x + 6
2
Find two numbers that multiply to 18 and add to 11: +9 and +2
3
Rewrite the middle term: 3x² + 9x + 2x + 6
4
Factorise in pairs: 3x(x + 3) + 2(x + 3) = (3x + 2)(x + 3)
Answer(3x + 2)(x + 3)
Difference of two squares
A special case: a² − b² = (a + b)(a − b). Recognising this pattern saves significant time in the exam.
Worked Example 5
Factorise: 9x² − 25
1
Recognise as difference of two squares: (3x)² − 5²
2
Apply the formula: (3x + 5)(3x − 5)
Answer(3x + 5)(3x − 5)
Completing the square
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. It is used to solve quadratics and find the vertex (minimum or maximum point) of a parabola.
For x² + bx + c: write (x + b/2)² − (b/2)² + c
Worked Example 6
Complete the square for: x² + 6x − 7
1
Half the coefficient of x: 6 ÷ 2 = 3
2
Write (x + 3)² − 3² − 7 = (x + 3)² − 9 − 7 = (x + 3)² − 16
Answer(x + 3)² − 16
When a quadratic cannot be easily factorised, use the formula:
x = (−b ± √(b² − 4ac)) / 2a
The expression b² − 4ac is called the discriminant. If it is negative, there are no real solutions. If it equals zero, there is exactly one solution.
Worked Example 7
Solve: 2x² − 5x − 3 = 0 using the quadratic formula
1
Identify a = 2, b = −5, c = −3
2
Calculate the discriminant: b² − 4ac = 25 + 24 = 49
3
Substitute: x = (5 ± √49) / 4 = (5 ± 7) / 4
4
Two solutions: x = 12/4 = 3 or x = −2/4 = −0.5
Answerx = 3 or x = −0.5
Practice questions
Work through each question before checking the answers. Difficulty is shown for each question.
Foundation (Grade 3–5)
Q1Expand and simplify: (x + 6)(x + 2)Foundation
Q2Expand and simplify: (x − 4)(x − 3)Foundation
Q3Factorise: x² + 7x + 10Foundation
Q4Factorise: x² − 9x + 18Foundation
Q5Solve by factorising: x² + 5x + 6 = 0Foundation
Higher (Grade 5–7)
Q6Expand and simplify: (2x + 1)(3x − 4)Higher
Q7Factorise: x² − 16Higher
Q8Factorise: 2x² + 7x + 3Higher
Q9Solve: x² − 2x − 8 = 0 by factorisingHigher
Q10Complete the square for: x² − 8x + 3Higher
Higher — Hard (Grade 8–9)
Q11Factorise: 6x² − 13x − 5Grade 8–9
Q12Solve using the quadratic formula (give answers to 2 d.p.): 3x² − 7x + 2 = 0Grade 8–9
Q13Complete the square for 2x² − 12x + 7 and hence find the minimum valueGrade 8–9
Q14Solve: x² − 6x + 4 = 0, leaving your answer in surd formGrade 8–9
Q15Show that the equation 4x² + 3x + 2 = 0 has no real solutionsGrade 8–9
Answers
Foundation (Q1–Q5)
Q1x² + 8x + 12(FOIL: x² + 2x + 6x + 12)
Q2x² − 7x + 12(FOIL: x² − 3x − 4x + 12)
Q3(x + 2)(x + 5)(two numbers: +2 and +5, multiply to 10, add to 7)
Q4(x − 3)(x − 6)(two numbers: −3 and −6, multiply to 18, add to −9)
Q5x = −2 or x = −3((x + 2)(x + 3) = 0)
Higher (Q6–Q10)
Q66x² − 5x − 4(FOIL: 6x² − 8x + 3x − 4)
Q7(x + 4)(x − 4)(difference of two squares: x² − 4²)
Q8(2x + 1)(x + 3)(ac = 6; numbers: +6 and +1; split: 2x² + 6x + x + 3)
Q9x = 4 or x = −2((x − 4)(x + 2) = 0)
Q10(x − 4)² − 13(half of −8 is −4; (x−4)² − 16 + 3)
Higher — Hard (Q11–Q15)
Q11(2x − 5)(3x + 1)(ac = −30; numbers: −15 and +2; split: 6x² − 15x + 2x − 5)
Q12x = 2.00 or x = 0.33(discriminant = 49 − 24 = 25; x = (7 ± 5) / 6)
Q132(x − 3)² − 11; minimum value = −11(factor 2 first: 2(x² − 6x) + 7; complete: 2(x−3)² − 18 + 7)
Q14x = 3 ± √5(quadratic formula: x = (6 ± √(36−16)) / 2 = (6 ± √20) / 2 = 3 ± √5)
Q15Discriminant = 9 − 32 = −23 < 0, so no real solutions(b² − 4ac = 3² − 4(4)(2))
Common mistakes
These are the errors Alamin sees most frequently with quadratic expressions and equations. Recognising them now will save you marks in the exam.
Common Mistake 1
Forgetting the middle terms when expanding
Students write (x + 3)² = x² + 9. This is wrong. The correct expansion is (x + 3)(x + 3) = x² + 6x + 9. Always use FOIL — the two middle terms are not zero.
Common Mistake 2
Wrong signs when factorising
For x² − 5x + 6, students often write (x + 2)(x + 3). This is wrong. Check: the numbers must add to −5 (both negative): (x − 2)(x − 3). Always verify by expanding back.
Common Mistake 3
Not setting each bracket to zero when solving
After factorising to (x − 2)(x + 5) = 0, students write x = 2 and x = 5. This is wrong. The second bracket gives x + 5 = 0, so x = −5, not +5.
Common Mistake 4
Arithmetic errors with the quadratic formula
The most common error is calculating b² − 4ac incorrectly, especially when b is negative. If b = −3, then b² = (−3)² = 9, not −9. Write out each substitution on its own line.
Common Mistake 5
Forgetting to subtract (b/2)² when completing the square
For x² + 8x + 3, writing (x + 4)² + 3 is wrong. Adding the bracket introduces an extra +16 that must be subtracted: (x + 4)² − 16 + 3 = (x + 4)² − 13.
Common Mistake 6
Incorrectly factorising the difference of two squares
Students write x² − 25 = (x − 5)(x − 5). This is wrong — that expands to (x−5)² = x² − 10x + 25. The correct factorisation is (x + 5)(x − 5), with one positive and one negative bracket.
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