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Completing the Square

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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 13 marks

Write 2x^2+6x-1 in the form a(x+b)^2+c

Give b and c as fractions in their simplest form.

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

Hint

Take the 2 out of the first two terms only, then halve the coefficient of x inside the bracket.

Worked solution
  1. Take out the 2 from the x terms: 2(x^2+3x)-1
  2. Complete the square inside: x^2+3x=\left(x+\frac32\right)^2-\frac94
  3. So 2\left[\left(x+\frac32\right)^2-\frac94\right]-1
  4. Multiply out the outer bracket: 2\left(x+\frac32\right)^2-\frac92-1
  5. -\frac92-1=-\frac{11}{2}
  6. Answer: 2\left(x+\frac32\right)^2-\frac{11}{2}

Question 22 marks

y=x^2+10x+k, where k is a constant.

The minimum value of y is 3

Work out the value of k.

Hint

Complete the square first; the number left outside the bracket is the minimum value of y.

Worked solution
  1. Complete the square: x^2+10x=(x+5)^2-25
  2. So y=(x+5)^2+k-25
  3. (x+5)^2\geqslant 0, so the minimum value of y is k-25
  4. k-25=3
  5. Answer: k=28

Question 3Challenge6 marks

A curve has equation y=x^2-2ax+b, where a and b are constants.

(a)

Write x^2-2ax+b in the form (x+p)^2+q, where p and q are in terms of a and b.

2 marks

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

(b)

The curve passes through the point (2,\ 7).

The minimum point of the curve lies on the line y=x-1

Work out the two possible values of a.

4 marks

Give every value, separated by commas

Hint

The minimum point of (x+p)^2+q is (-p,\ q); use the point on the curve to replace b, then substitute the minimum point into the line.

Worked solution

Part (a)

  1. Halve the coefficient of x: -2a\div2=-a
  2. x^2-2ax=(x-a)^2-a^2
  3. Answer: (x-a)^2+b-a^2

Part (b)

  1. Substitute (2,\ 7) into the curve: 4-4a+b=7, so b=4a+3
  2. From part (a), the minimum point is (a,\ b-a^2)
  3. So the minimum point is (a,\ 4a+3-a^2)
  4. It lies on y=x-1: 4a+3-a^2=a-1
  5. Rearrange: a^2-3a-4=0
  6. Factorise: (a-4)(a+1)=0
  7. Answer: a=4 or a=-1

More on this topic: Completing the Square worksheet with full solutions

All AQA Level 2 Further Maths practice questions

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