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Identities and Comparing Coefficients

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 12 marks

4(3x-k)+2(x+5)\equiv 14x-6 Work out the value of k.

Hint

Expand the left-hand side, then compare the constant terms on each side.

Worked solution
  1. Expand: 12x-4k+2x+10
  2. Collect like terms: 14x+(10-4k)
  3. The x terms already match: 14x
  4. Compare the constant terms: 10-4k=-6
  5. -4k=-16
  6. k=4

Question 22 marks

(x+k)^3\equiv x^3+ax^2+bx-64 where k, a and b are constants.

Work out the value of b.

Hint

Start with the constant term: what must k^3 equal?

Worked solution
  1. The constant term of (x+k)^3 is k^3
  2. So k^3=-64, which gives k=-4
  3. (x-4)^2=x^2-8x+16
  4. (x-4)^3=(x-4)(x^2-8x+16)=x^3-12x^2+48x-64
  5. Compare the x terms: b=48

Question 3Challenge5 marks

(x+a)^2-b(x+3)\equiv x^2+2x-2 where a and b are constants.

(a)

By comparing the coefficients of x, write b in terms of a.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Work out the two possible values of a.

Do not use trial and improvement.

2 marks

Give every value, separated by commas

(c)

Work out the value of b when a takes its larger value.

1 mark

Hint

Expand the left-hand side fully and collect the x terms and the constant terms before comparing with the right-hand side.

Worked solution

Part (a)

  1. Expand: (x+a)^2-b(x+3)=x^2+2ax+a^2-bx-3b
  2. The x terms are (2a-b)x
  3. Compare with 2x: 2a-b=2
  4. b=2a-2

Part (b)

  1. Compare the constant terms: a^2-3b=-2
  2. Substitute b=2a-2 (use a bracket): a^2-3(2a-2)=-2
  3. a^2-6a+6=-2
  4. a^2-6a+8=0
  5. (a-2)(a-4)=0
  6. a=2 or a=4

Part (c)

  1. a=4, so b=2(4)-2=6
  2. Check: (x+4)^2-6(x+3)=x^2+8x+16-6x-18=x^2+2x-2 ✓

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