[discia] GCSE Maths › Algebra › Completing the Square

Completing the Square

discia.org 10 questions · 30 marks
Name Date
  1. Either paper 3 marks

    For all values of \(x\),

    \(x^2 + bx + c = (x - 9)^2 + 7\)

    (a)

    Write down the value of \(b\).

    (b)

    Calculate the value of \(c\).

    Show answer
    (a) \(-18\)
    (b) \(88\)
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  2. Either paper Higher2 marks

    Complete the square for \(x^2 + 18x + 86\).

    Give your answer in the form \((x + p)^2 + q\).

    Show answer
    \((x + 9)^2 + 5\)
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  3. Either paper Higher3 marks

    The curve with equation

    \(y = x^2 + bx + c\)

    has a turning point at \((7, -2)\).

    Find the value of \(b\) and the value of \(c\).

    Show answer
    \(b = -14\) and \(c = 47\)
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  4. Either paper Higher2 marks

    A student was asked to write \(x^2 - 16x + 76\) in the form \((x + p)^2 + q\).

    Here is the student's answer.

    \((x - 8)^2 + 76\)

    The student has the right number inside the bracket, but the answer is wrong.

    (a)

    Explain the mistake the student has made.

    (b)

    State the correct value of \(q\).

    Show answer
    (a) Squaring \((x - 8)\) produces an extra \(64\) that the original expression does not have, and the student has not subtracted it again — the number on the end has simply been copied.
    (b) \(q = 12\)
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  5. Either paper Higher4 marks
    (a)

    Write \(x^2 - 10x + 26\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are integers.

    (b)

    Hence prove that \(x^2 - 10x + 26\) is always positive.

    Show answer
    (a) \((x - 5)^2 + 1\)
    (b) \((x - 5)^2\) is a square, so it is never negative. The smallest the whole expression can be is \(1\), which is greater than zero, so \(x^2 - 10x + 26\) is always positive.
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  6. Either paper Higher4 marks

    A curve has equation \(y = x^2 - 2x + 21\)

    (a)

    Write the right-hand side of this equation in the form \((x + p)^2 + q\).

    (b)

    Write down the coordinates of the turning point of the curve.

    (c)

    Write down the equation of the line of symmetry of the curve.

    Show answer
    (a) \((x - 1)^2 + 20\)
    (b) \((1, 20)\)
    (c) \(x = 1\)
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  7. Either paper Higher3 marks

    Solve \(x^2 - 12x + 29 = 0\) by completing the square.

    Give your answers in surd form.

    Show answer
    \(x = 6 \pm \sqrt{ 7 }\)
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  8. Either paper Higher3 marks

    Complete the square for \(4x^2 + 32x + 87\).

    Give your answer in the form \(a(x + p)^2 + q\).

    Show answer
    \(4(x + 4)^2 + 23\)
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  9. Either paper Higher2 marks

    The coefficient of \(x\) in this expression is odd, so \(p\) and \(q\) will both be fractions.

    \(x^2 + 11x - 10\)

    Write the expression in the form \((x + p)^2 + q\).

    Show answer
    \((x + \frac{11}{2})^2 - \frac{161}{4}\)
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  10. Either paper Higher4 marks

    Here are two expressions.

    \(A = x^2 - 10x + 12\)

    \(B = x^2 + 8x + 15\)

    Which of the two expressions has the smaller minimum value?

    You must show your working.

    Show answer
    \(A\). The minimum value of \(A\) is \(-13\) and the minimum value of \(B\) is \(-1\).
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discia.org Completing the Square · sheet 1
Answers Completing the Square · sheet 1
  1. (a) \(-18\)
    (b) \(88\)
  2. \((x + 9)^2 + 5\)
  3. \(b = -14\) and \(c = 47\)
  4. (a) Squaring \((x - 8)\) produces an extra \(64\) that the original expression does not have, and the student has not subtracted it again — the number on the end has simply been copied.
    (b) \(q = 12\)
  5. (a) \((x - 5)^2 + 1\)
    (b) \((x - 5)^2\) is a square, so it is never negative. The smallest the whole expression can be is \(1\), which is greater than zero, so \(x^2 - 10x + 26\) is always positive.
  6. (a) \((x - 1)^2 + 20\)
    (b) \((1, 20)\)
    (c) \(x = 1\)
  7. \(x = 6 \pm \sqrt{ 7 }\)
  8. \(4(x + 4)^2 + 23\)
  9. \((x + \frac{11}{2})^2 - \frac{161}{4}\)
  10. \(A\). The minimum value of \(A\) is \(-13\) and the minimum value of \(B\) is \(-1\).
discia.org Completing the Square · sheet 1