Completing the Square worksheet
10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.
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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 \(c\).
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2 marks
Complete the square for \(x^2 + 18x + 86\).
Give your answer in the form \((x + p)^2 + q\).
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3 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 \(c\).
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2 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 \(q\).
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4 marks
A curve has equation \(y = x^2 - 10x + 1\)
(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.
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3 marks
\(x^2 - 10x + 3 = 0\)
Work out the values of \(x\) by completing the square.
Give your answers in surd form.
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3 marks
Write \(2x^2 - 8x + 31\) in the form \(a(x + p)^2 + q\).
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2 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 \((x + p)^2 + q\).
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4 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.
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4 marks
Here is a quadratic expression.
\(38 - 8x - x^2\)
(a)Write the expression in the form \(q - (x + p)^2\).
(b)State the maximum value of the expression.
(c)State the value of \(x\) at which that maximum occurs.
Answers
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(a) \(-18\)
(b) \(88\) - \((x + 9)^2 + 5\)
- \(b = -14\) and \(c = 47\)
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(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) \(q = 12\) -
(a) \((x - 5)^2 - 24\)
(b) \((5, -24)\)
(c) \(x = 5\) - \(x = 5 \pm \sqrt{ 22 }\)
- \(2(x - 2)^2 + 23\)
- \((x + \frac{11}{2})^2 - \frac{161}{4}\)
- \(A\). The minimum value of \(A\) is \(-13\) and the minimum value of \(B\) is \(-1\).
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(a) \(54 - (x + 4)^2\)
(b) \(54\)
(c) \(-4\)