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Harder factorising (ax² + bx + c) practice questions with answers

Higher · Grade 7 · Non-calculator

A second set of harder factorising (ax² + bx + c) 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

Factorise 3x²+10x+3.

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Answer: (3x+1)(x+3)

  1. Test splits of 3 and 3.
  2. Cross terms 9x + x = 10x: (3x+1)(x+3).
  • Method (1): Test splits of 3 and 3.
  • Accuracy (1): Cross terms 9x + x = 10x: (3x+1)(x+3).

Question 2 Easy

Factorise 2x²+9x+4.

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Answer: (2x+1)(x+4)

  1. Test splits of 2 and 4.
  2. Cross terms 8x + x = 9x: (2x+1)(x+4).
  • Method (1): Test splits of 2 and 4.
  • Accuracy (1): Cross terms 8x + x = 9x: (2x+1)(x+4).

Question 3 Medium

Factorise 6x²+11x+3.

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Answer: (3x+1)(2x+3)

  1. Split 6 as 3 and 2, and 3 as 1 and 3.
  2. Cross terms 9x + 2x = 11x: (3x+1)(2x+3).
  • Method (1): Split 6 as 3 and 2, and 3 as 1 and 3.
  • Accuracy (1): Cross terms 9x + 2x = 11x: (3x+1)(2x+3).

Question 4 Medium

Factorise 4x²+12x+5.

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Answer: (2x+1)(2x+5)

  1. Split 4 as 2 and 2, and 5 as 1 and 5.
  2. Cross terms 10x + 2x = 12x: (2x+1)(2x+5).
  • Method (1): Split 4 as 2 and 2, and 5 as 1 and 5.
  • Accuracy (1): Cross terms 10x + 2x = 12x: (2x+1)(2x+5).

Question 5 Hard

Solve 2x²+5x+3=0.

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Answer: -3/2 or -1

  1. Factorise: (2x+3)(x+1) = 0.
  2. Roots are x = -3/2 or x = -1.
  • Method (1): Factorise: (2x+3)(x+1) = 0.
  • Accuracy (1): Roots are x = -3/2 or x = -1.

Question 6 Hard

Solve 3x²+10x+3=0.

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Answer: -3 or -1/3

  1. Factorise: (3x+1)(x+3) = 0.
  2. Roots are x = -3 or x = -1/3.
  • Method (1): Factorise: (3x+1)(x+3) = 0.
  • Accuracy (1): Roots are x = -3 or x = -1/3.

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 13 marks

Factorise 3x² + 8x + 4.

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

  1. Split 3x² as 3x and x: (3x + ?)(x + ?).
  2. Test factors of 4: 2 and 2 give cross terms 3x(2) + 2(x) = 8x.
  3. Form factorisation: (3x + 2)(x + 2).
  4. Check: 3x² + 6x + 2x + 4 = 3x² + 8x + 4.
  • Method (1): (3x + a)(x + b) with ab = 4
  • Method (1): Cross terms that add to 8x
  • Accuracy (1): (3x+2)(x+2)

Quick checks

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

Check 1

How do you factorise 2x²+7x+3?

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

(2x+1)(x+3) Brackets are (2x+1)(x+3).

Check 2

Write the brackets for 6x²+5x+1.

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

(3x+1)(2x+1) Brackets are (3x+1)(2x+1).

Check 3

Give the factorised form of 3x²+10x+3.

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

(3x+1)(x+3) Brackets are (3x+1)(x+3).

Check 4

Factorise the difference of squares 4x²-9.

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(2x-3)(2x+3). (2x-3)(2x+3).

Check 5

Factorise the difference of squares 9x²-16.

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(3x-4)(3x+4). (3x-4)(3x+4).

Check 6

Why expand the brackets to check?

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The cross terms must sum to the original middle term. The cross terms must sum to the original middle term.

Check 7

Solve 2x²+7x+3 = 0.

  1. -3 or -1/2
  2. -3 or -1/3
  3. middle term is 5x, not 7x
Show answer

-3 or -1/2 Roots are x = -3 or x = -1/2.

Check 8

Solve 3x²+10x+3 = 0.

  1. -3 or -1/3
  2. middle term is 5x, not 7x
  3. -3 or -1/2
Show answer

-3 or -1/3 Roots are x = -3 or x = -1/3.

Check 9

Factorise 2x²+5x+3.

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

(2x+3)(x+1) Brackets are (2x+3)(x+1).

Check 10

Factorise 3x²+7x+2.

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

(3x+1)(x+2) Brackets are (3x+1)(x+2).

Check 11

When solving (2x+1)(x+3) = 0, what does 2x+1 = 0 give?

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Subtract 1 and divide by 2: x = -1/2. Subtract 1 and divide by 2: x = -1/2.

Check 12

When solving (3x+1)(x+3) = 0, what does 3x+1 = 0 give?

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Subtract 1 and divide by 3: x = -1/3. Subtract 1 and divide by 3: x = -1/3.

Check 13

Why is (2x+3)(x+1) wrong for 2x²+7x+3?

  1. -3 or -1/3
  2. middle term is 5x, not 7x
  3. -3 or -1/2
Show answer

middle term is 5x, not 7x Expanding gives 5x instead of 7x.

Check 14

How does 4x² - 9 factorise as difference of squares?

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Square roots are 2x and 3. Square roots are 2x and 3.

Check 15

What signs appear in the brackets for a difference of two squares?

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Difference of two squares always produces one plus bracket and one minus bracket. Difference of two squares always produces one plus bracket and one minus bracket.

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