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Standard Level

1.02 Solve quadratics by completing the square

Worksheet
Completing the square
1

Explain how to find the number to add to x^{2} - 5 x to make it a perfect square trinomial.

2

Consider the expression x^{2} + 6 x.

a

Does adding 36 to the expression make it a perfect square?

b

What should be added to x^{2} + 6 x to make it a perfect square?

3

Consider the equation x^{2} - 3 x = 6. What should be added to the equation in order to complete the square?

4

Complete the following expressions so they form a perfect square:

a

x^{2} + 10 x+⬚

b

x^{2} - 2 x + ⬚

c

x^{2}-⬚x+16

d

x^{2} - x + ⬚

e

x^{2} + m x + ⬚

f

x^{2} - \dfrac{4}{5} x + ⬚

5

Complete the following perfect squares:

a

x^{2} + 4 x + ⬚ = \left(x + ⬚\right)^{2}

b

x^{2} - 5 x + ⬚ = \left(x - ⬚\right)^{2}

c

x^{2}-\dfrac{7}{4} x+⬚=(x-⬚)^2

6

Rewrite the following in the form \left(x + b\right)^{2} + c using the method of completing the square:

a

x^{2} + 12 x

b

x^{2} - 12 x

c

x^{2} + 2 x + 2

d

x^{2} - 14 x + 56

e

x^{2} - 6 x + 5

f

x^{2} + 3 x + 6

g

x^{2} - 7 x + 14

7

Factorise the quadratics using completing the square:

a

y = x^{2} + 56 x + 159

b

y = x^{2} - 24 x + 63

8

Factorise the quadratics using the method of completing the square to get them into the form:y = \left(x - a + \sqrt{b}\right) \left(x - a - \sqrt{b}\right)

a

y = x^{2} - 6 x + 7

b

y = x^{2} + 8 x + 14

Non-monic completing the square
9

Using the method of completing the square, rewrite 4 x^{2} - 36 x + 79 in the form c\left(\left(x + a\right)^2 + b\right).

10

Rewrite the following in the form a\left(x + b\right)^2 + c using the method of completing the square:

a

3 x^{2} + 9 x + 8

b

4 x^{2} - 11 x + 7

11

Use completing the square to factorise the quadratic y = 2 x^{2} + 25 x + 23.

Solving equations
12

Consider the equation x^{2} + 24 x = 10. To solve the equation by completing the square, state whether each of the following lines of working could be used:

a
x^{2} + 24 x + 144 = 10
b
\left(x + 12\right)^{2} - 144 = 10
c
\left(x + 12\right)^{2} = 10
d
\left(x + 12\right)^{2} - 144 = - 134
e
x^{2} + 24 x + 144 = 10 + 144
13

Solve the following quadratic equations:

a

\left(x + 6\right)^{2} = 4

b

\left( 4 x - 4\right)^{2} = 4

c

2 \left(x - 3\right)^{2} = 8

d

\left(x + 2\right)^{2} = 20

e

\left(x + 5\right)^{2} - 2 = 15

f

\left(x + 9\right)^{2} = \dfrac{15}{2}

14

Solve the following quadratic equations by completing the square:

a

x^{2} + 18 x + 32 = 0

b

x^{2} - 6 x + 8 = 0

c

x^{2} - 9 x + 8 = 0

d

2 x^{2} - 12 x - 32 = 0

e

4 x^{2} + 11 x + 7 = 0

f

x^{2} - 2 x - 32 = 0

15

Solve the following quadratic equations by completing the square. Leave your answers in surd form:

a

x^{2} + 24 x + 5 = 0

b

x^{2} + 11 x + 5 = 0

c

6 x^{2} + 48 x + 24 = 0

d

x^{2} - 7 x + 8 = 0

e

x^{2} + \dfrac{x}{3} - 3 = 0

f

5 x^{2} + 55 x + 3 = 0

Applications
16

A circle has equation \left(x + 2\right)^{2} + \left(y - 4\right)^{2} = 41. Find the x-intercepts of the circle.

17

Find the value(s) of k that will make x^{2} + k x + 16 a perfect square trinomial.

18

The revenue y (in millions of dollars) of a company x years after it first started is modelled by y = 12.5 x^{2} - 64 x + 135

By completing the square, use this equation to predict the number of years x after which the revenue of the company will reach \$1167 million.

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