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GCSE Edexcel Maths › Completing the Square(GCSE Maths)

Completing the Square(GCSE Maths)

Skill Check

Q1
Question 1

Solve the equation (use completing the square): x² − 8x + 5 = 0.

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SOLUTION

Step 1  ·  Complete the square

$$(x - 4)^2 - 16 + 5 = 0$$

Step 2  ·  Simplify the equation

$$(x - 4)^2 - 11 = 0$$

Step 3  ·  Rearrange the equation

$$(x - 4)^2 = 11$$

Step 4  ·  Square root both sides

$$x - 4 = \pm \sqrt{11}$$

Step 5  ·  Solve for $x$

$$x = 4 \pm \sqrt{11}$$

Final Answer

$x = 4 \pm \sqrt{11}$

Q2
Question 2

Solve the equation (use completing the square): x² + 6x − 16 = 0.

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SOLUTION

Step 1  ·  Complete the square

$$(x + 3)^2 - 9 - 16 = 0$$

Step 2  ·  Simplify the equation

$$(x + 3)^2 - 25 = 0$$

Step 3  ·  Rearrange to isolate the squared term

$$(x + 3)^2 = 25$$

Step 4  ·  Square root both sides

$$x + 3 = \pm 5$$

Step 5  ·  Solve for $x$

$$x = -3 + 5 = 2$$
$$x = -3 - 5 = -8$$

Final Answer

$x = 2 \text{ or } x = -8$

Q3
Question 3

Write the expression x² + 5x + 2 in the form (x + a)² + b.

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SOLUTION

Step 1  ·  Have the $x$ coefficient

Start by writing the squared term using half of the $x$ coefficient:

$$(x + 2.5)^2$$

Step 2  ·  Subtract the square

Subtract the square of $2.5$ from the expression and add the original constant:

$$(x + 2.5)^2 - 6.25 + 2$$

Step 3  ·  Simplify the constant terms

Combine $-6.25$ and $2$ to simplify the expression completely:

$$(x + 2.5)^2 - 4.25$$

Alternatively, this can be written using fractions:

$$\left(x + \frac{5}{2}\right)^2 - \frac{17}{4}$$

Final Answer

$(x + 2.5)^2 - 4.25$

Q4
Question 4

Write the expression x² − 14x − 3 in the form (x + a)² + b.

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SOLUTION

Step 1  ·  Have the $x$ coefficient

$$(x - 7)^2$$

Step 2  ·  Subtract the square of $-7$

$$(x - 7)^2 - 49 - 3$$

Step 3  ·  Simplify the expression

$$(x - 7)^2 - 52$$

Final Answer

$(x - 7)^2 - 52$

Q5
Question 5

Write the expression x² + 12x + 5 in the form (x + a)² + b.

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SOLUTION

Step 1  ·  Have the $x$ coefficient

$$(x + 6)^2$$

Step 2  ·  Subtract the square of 6

$$(x + 6)^2 - 36 + 5$$

Step 3  ·  Simplify the expression

$$(x + 6)^2 - 31$$

Final Answer

$(x + 6)^2 - 31$

Problem Solving

Q6
Question 6

Solve the equation (use completing the square): 4x² + 8x − 21 = 0

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SOLUTION

Step 1  ·  Factor out the coefficient

$$4(x^2 + 2x) - 21 = 0$$

Step 2  ·  Complete the square

$$4[(x + 1)^2 - 1] - 21 = 0$$

Step 3  ·  Expand and simplify

$$4(x + 1)^2 - 4 - 21 = 0$$
$$4(x + 1)^2 = 25$$

Step 4  ·  Isolate the squared term and root

$$(x + 1)^2 = 6.25$$
$$x + 1 = \pm 2.5$$

Step 5  ·  Solve for $x$

$$x = 1.5 \quad \text{or} \quad x = -3.5$$

Final Answer

$x = 1.5 \quad \text{or} \quad x = -3.5$

Q7
Question 7

Solve the equation (use completing the square): 2x² − 12x + 5 = 0

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SOLUTION

Step 1  ·  Factor out 2 from the $x$ terms

$$2(x^2 - 6x) + 5 = 0$$

Step 2  ·  Complete the square inside the bracket

$$2[(x - 3)^2 - 9] + 5 = 0$$

Step 3  ·  Expand and simplify

Multiply out the square bracket and collect the constant terms:

$$2(x - 3)^2 - 18 + 5 = 0$$
$$2(x - 3)^2 = 13$$

Step 4  ·  Isolate $x$ to solve the equation

Divide both sides by $2$ and take the square root:

$$(x - 3)^2 = 6.5$$
$$x - 3 = \pm \sqrt{6.5}$$
$$x = 3 \pm \sqrt{6.5}$$

Final Answer

$x = 3 \pm \sqrt{6.5}$

Q8
Question 8

Write the expression 5x² + 30x + 11 in the form a(x + b)² + c.

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SOLUTION

Step 1  ·  Factor out $5$

$$5(x^2 + 6x) + 11$$

Step 2  ·  Complete the square

$$5[(x + 3)^2 - 9] + 11$$

Step 3  ·  Expand the brackets

$$5(x + 3)^2 - 45 + 11$$

Step 4  ·  Simplify the constant terms

$$5(x + 3)^2 - 34$$

Final Answer

$5(x + 3)^2 - 34$

Q9
Question 9

Write the expression 3x² − 12x − 10 in the form a(x + b)² + c.

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SOLUTION

Step 1  ·  Factor out 3

$$3(x^2 - 4x) - 10$$

Step 2  ·  Complete the square

$$3[(x - 2)^2 - 4] - 10$$

Step 3  ·  Expand the bracket

$$3(x - 2)^2 - 12 - 10$$

Step 4  ·  Simplify the expression

$$3(x - 2)^2 - 22$$

Final Answer

$3(x - 2)^2 - 22$

Q10
Question 10

Write the expression 2x² + 16x + 7 in the form a(x + b)² + c.

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SOLUTION

Step 1  ·  Factor out 2 from the $x$ terms

$$2(x^2 + 8x) + 7$$

Step 2  ·  Complete the square inside the bracket

$$2[(x + 4)^2 - 16] + 7$$

Step 3  ·  Expand the outer bracket

$$2(x + 4)^2 - 32 + 7$$

Step 4  ·  Simplify the expression

$$2(x + 4)^2 - 25$$

Final Answer

$2(x + 4)^2 - 25$

Exam-Style Questions

Q11
Question 11

Solve the equation (use completing the square): 12x² − 24x − 11 = 0.

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SOLUTION

Step 1  ·  Factor out 12

$$12(x^2 - 2x) - 11 = 0$$

Step 2  ·  Complete the square

$$12[(x - 1)^2 - 1] - 11 = 0$$

Step 3  ·  Expand and simplify

$$12(x - 1)^2 - 12 - 11 = 0$$
$$12(x - 1)^2 = 23$$

Step 4  ·  Solve for $x$

$$(x - 1)^2 = \frac{23}{12}$$
$$x - 1 = \pm \sqrt{\frac{23}{12}}$$
$$x = 1 \pm \sqrt{\frac{23}{12}}$$

Final Answer

$x = 1 \pm \sqrt{\frac{23}{12}}$

Q12
Question 12

Solve the equation (use completing the square): 10x² + 40x − 17 = 3.

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SOLUTION

Step 1  ·  Set the equation to zero

$$10x^2 + 40x - 20 = 0$$

Step 2  ·  Factor out the coefficient

Factor out $10$ from the $x$ terms:

$$10(x^2 + 4x) - 20 = 0$$

Step 3  ·  Complete the square

Complete the square for the expression inside the brackets:

$$10[(x + 2)^2 - 4] - 20 = 0$$

Step 4  ·  Expand and simplify

Expand the brackets and move the constant terms to the right-hand side:

$$10(x + 2)^2 - 40 - 20 = 0$$
$$10(x + 2)^2 = 60$$

Step 5  ·  Solve for $x$

Divide by $10$ and square root both sides:

$$(x + 2)^2 = 6$$
$$x + 2 = \pm \sqrt{6}$$
$$x = -2 \pm \sqrt{6}$$

Final Answer

$x = -2 \pm \sqrt{6}$

Q13
Question 13

Write the expression −2x² + 16x − 5 in the form a(x + b)² + c.

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SOLUTION

Step 1  ·  Factor out $-2$

$$-2(x^2 - 8x) - 5$$

Step 2  ·  Complete the square

$$-2[(x - 4)^2 - 16] - 5$$

Step 3  ·  Expand and watch the negatives

$$-2(x - 4)^2 + 32 - 5$$

Step 4  ·  Simplify

$$-2(x - 4)^2 + 27$$

Final Answer

$-2(x - 4)^2 + 27$

Q14
Question 14

Write the expression 12x² − 36x + 17 in the form a(x + b)² + c.

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SOLUTION

Step 1  ·  Factor out the coefficient

Factor out 12 from the $x$ terms:

$$12(x^2 - 3x) + 17$$

Step 2  ·  Complete the square

Complete the square for the expression inside the bracket:

$$12[(x - 1.5)^2 - 2.25] + 17$$

Step 3  ·  Expand the bracket

Multiply the terms inside the square bracket by 12:

$$12(x - 1.5)^2 - 27 + 17$$

Step 4  ·  Simplify the expression

Combine the constant terms:

$$12(x - 1.5)^2 - 10$$

Final Answer

$12(x - 1.5)^2 - 10$

Q15
Question 15

Write the expression 12x² + 48x − 5 in the form a(x + b)² + c.

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SOLUTION

Step 1  ·  Factor out 12

$$12(x^2 + 4x) - 5$$

Step 2  ·  Complete the square

$$12[(x + 2)^2 - 4] - 5$$

Step 3  ·  Expand the brackets

$$12(x + 2)^2 - 48 - 5$$

Step 4  ·  Simplify

$$12(x + 2)^2 - 53$$

Final Answer

$12(x + 2)^2 - 53$

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