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Expanding & factorising quadratics practice questions with answers

Foundation and Higher · Grade 5 · Non-calculator

A second set of expanding & factorising quadratics questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

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Question 1 Easy

Expand (x+6)(x+1).

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Answer: x^2+7x+6

  1. Multiply out: x^2 + x + 6x + 6.
  2. Collect terms: x^2 + 7x + 6.
  • Method (1): Multiply out: x^2 + x + 6x + 6.
  • Accuracy (1): Collect terms: x^2 + 7x + 6.

Question 2 Easy

Expand (x+2)(x+2).

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Answer: x^2+4x+4

  1. Multiply out: x^2 + 2x + 2x + 4.
  2. Collect terms: x^2 + 4x + 4.
  • Method (1): Multiply out: x^2 + 2x + 2x + 4.
  • Accuracy (1): Collect terms: x^2 + 4x + 4.

Question 3 Medium

Factorise x^2+5x+6.

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Answer: (x+2)(x+3)

  1. Find pair: 2 and 3 sum to 5 and multiply to 6.
  2. Write brackets: (x+2)(x+3).
  • Method (1): Find pair: 2 and 3 sum to 5 and multiply to 6.
  • Accuracy (1): Write brackets: (x+2)(x+3).

Question 4 Medium

Factorise x^2+6x+8.

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Answer: (x+2)(x+4)

  1. Find pair: 2 and 4 sum to 6 and multiply to 8.
  2. Write brackets: (x+2)(x+4).
  • Method (1): Find pair: 2 and 4 sum to 6 and multiply to 8.
  • Accuracy (1): Write brackets: (x+2)(x+4).

Question 5 Hard

Factorise x^2-5x+6.

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Answer: (x-3)(x-2)

  1. Find pair: -3 and -2 sum to -5 and multiply to 6.
  2. Write brackets: (x-3)(x-2).
  • Method (1): Find pair: -3 and -2 sum to -5 and multiply to 6.
  • Accuracy (1): Write brackets: (x-3)(x-2).

Question 6 Hard

Factorise 2a+2b.

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Answer: 2(a+b)

  1. Identify common factor 2.
  2. Factorise: 2(a+b).
  • Method (1): Identify common factor 2.
  • Accuracy (1): Factorise: 2(a+b).

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Factorise x^2-36.

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Answer: (x-6)(x+6)

  1. This is a difference of two squares: 36 = 6^2.
  2. Write the factors with opposite signs: (x-6)(x+6).
  • Method (1): (x - 6)(x + 6) with one sign wrong, or (x + 6)(x + 6)
  • Accuracy (1): (x - 6)(x + 6)

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

When expanding (x+a)(x+b), how is the x term formed?

  1. Add the two numbers: (a+b)x
  2. Multiply the two numbers: ab x
  3. Square the first term
Show answer

Add the two numbers: (a+b)x The inner and outer products ax + bx collect to (a+b)x.

Check 2

What is the expansion of (x+3)(x+2)?

  1. x^2+5x+6
  2. x^2+6
  3. x^2+6x+5
Show answer

x^2+5x+6 x^2 + (3+2)x + 3×2 = x^2 + 5x + 6.

Check 3

What is the expansion of (x+4)(x+1)?

  1. x^2+5x+4
  2. x^2+4
  3. x^2+4x+4
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x^2+5x+4 x^2 + (4+1)x + 4×1 = x^2 + 5x + 4.

Check 4

What is the middle term of (x+4)^2?

  1. 8x
  2. 4x
  3. 16x
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8x Twice the product: 2 × 4 × x = 8x.

Check 5

What is the expansion of (x+3)^2?

  1. x^2+6x+9
  2. x^2+9
  3. x^2+3x+9
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x^2+6x+9 x^2 + 2(3)x + 3^2 = x^2 + 6x + 9.

Check 6

What is the factorised form of x^2+5x+6?

  1. (x+2)(x+3)
  2. (x+1)(x+6)
  3. (x-2)(x-3)
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(x+2)(x+3) 2 and 3 add to 5 and multiply to 6.

Check 7

What is the factorised form of x^2+6x+8?

  1. (x+2)(x+4)
  2. (x+1)(x+8)
  3. (x-2)(x-4)
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(x+2)(x+4) 2 and 4 add to 6 and multiply to 8.

Check 8

What is the factorised form of x^2-9?

  1. (x-3)(x+3)
  2. (x-9)(x+1)
  3. (x-3)^2
Show answer

(x-3)(x+3) Difference of squares: √9 = 3.

Check 9

What is the factorised form of x^2-16?

  1. (x-4)(x+4)
  2. (x-16)(x+1)
  3. (x-4)^2
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(x-4)(x+4) Difference of squares: √16 = 4.

Check 10

What is the common factor in 2a + 2b?

  1. 2
  2. a
  3. b
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2 2 is common to both terms: 2(a+b).

Check 11

What is the middle term of (x+n)^2?

  1. 2nx
  2. nx
  3. n^2 x
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2nx The expansion gives nx + nx = 2nx.

Check 12

Why are signs opposite in a difference of squares?

  1. So the middle x terms cancel out
  2. Because square roots are negative
  3. To make both brackets positive
Show answer

So the middle x terms cancel out +bx and -bx cancel to 0.

Check 13

How can you check if your factorisation is correct?

  1. Expand the brackets again
  2. Divide by x
  3. Check the signs only
Show answer

Expand the brackets again Expanding should return the original quadratic expression.

Check 14

What is the factorised form of x^2-5x+6?

  1. (x-3)(x-2)
  2. (x+3)(x+2)
  3. (x-6)(x+1)
Show answer

(x-3)(x-2) -3 and -2 add to -5 and multiply to +6.

Check 15

What is the expansion of (x+5)^2?

  1. x^2+10x+25
  2. x^2+25
  3. x^2+5x+25
Show answer

x^2+10x+25 x^2 + 2(5)x + 25 = x^2 + 10x + 25.

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