Completing the square worksheets with answers

Cambridge IGCSE 0580, Extended tier

Write a quadratic in completed-square form, find its turning point and solve it exactly.

Sample sheet

Completing the square, Standard

Name Date

1
Solve $x^{2} - 5x + 4 = 0$ by completing the square.
[4]
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2
Solve $x^{2} - 12x + 2 = 0$ by completing the square. Give your answers in surd form.
[4]
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3
Write $x^{2} + 6x + 15$ in the form $(x + p)^{2} + q$ . Hence write down the coordinates of the turning point of the curve $y = x^{2} + 6x + 15$ .
[4]
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4
Write $x^{2} - 9x - 19$ in the form $(x + p)^{2} + q$ . Hence write down the coordinates of the turning point of the curve $y = x^{2} - 9x - 19$ .
[4]
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5
Write $x^{2} + 3x - 20$ in the form $(x + p)^{2} + q$ .
[3]
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6
Solve $x^{2} - 8x - 17 = 0$ by completing the square. Give your answers in surd form.
[4]
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Answers

Completing the square, Standard

1 $x = 1 \text{ or } x = 4$
2 $x = 6 + \sqrt{34} \text{ or } x = 6 - \sqrt{34}$
3 $(x + 3)^{2} + 6,\ (-3, 6)$
4 $(x - \frac{9}{2})^{2} - \frac{157}{4},\ (\frac{9}{2}, -\frac{157}{4})$
5 $(x + \frac{3}{2})^{2} - \frac{89}{4}$
6 $x = 4 + \sqrt{33} \text{ or } x = 4 - \sqrt{33}$

Method in brief

Halve the coefficient of x to get the number in the bracket, square the bracket, then correct the constant. The bracket is zero at the turning point, and taking square roots of both sides solves the equation.

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Answers

Completing the square, Standard

x = 1 \text{ or } x = 4

x = 6 + \sqrt{34} \text{ or } x = 6 - \sqrt{34}

(x + 3)^{2} + 6,\ (-3, 6)

(x - \frac{9}{2})^{2} - \frac{157}{4},\ (\frac{9}{2}, -\frac{157}{4})

(x + \frac{3}{2})^{2} - \frac{89}{4}

x = 4 + \sqrt{33} \text{ or } x = 4 - \sqrt{33}