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Stage 5.1-3

2.13 Completing the square

Worksheet
Monic complete the square
1

Find the missing coefficient or term so that the following expressions form a perfect square.

a
x^{2}-⬚ x + 81
b

x^{2} + 10 x+⬚

c

x^{2}-⬚x+16

d

x^{2}-⬚x+121

2

For each of the following expressions, determine the value of k to make the expression a perfect square:

a
x^{2} + x+k
b
x^{2} - 2 x+k
c
x^{2} + 19 x+k
d
x^{2} -\dfrac{4}{5} x+k
3

Complete the following perfect squares:

a

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

b

\left(x - ⬚\right)^{2} = x^{2} - \dfrac{4}{3} x + ⬚

c

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

d

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

e

x^{2}-\dfrac{7}{4} x+⬚=\left(x-⬚\right)^2

f

\left(x - ⬚\right)^{2} = x^{2} - \dfrac{3}{2} x + ⬚

4

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

a

x^{2} + 18 x

b

x^{2} - 8 x

c

x^{2} + 10 x + 31

d

x^{2} + 14 x + 47

e

x^{2} - 10 x + 30

f

x^{2} - 18 x + 77

g

x^{2} + 9 x + 16

h

x^{2} - 7 x + 15

5

Factorise the following quadratics using the method of completing the square:

a
x^{2} + 6 x + 4
b
x^{2} - 8 x + 11
c

x^{2} + 24 x + 63

d

x^{2} - 20 x + 19

e

x^{2} + 42 x + 185

f
x^{2} - 6 x + 5
g

x^{2} - 28 x + 115

h

x^{2} + 11 x + 10

i

x^{2} - 11 x + 30

j

\left(x + 3\right) \left(x + 19\right) - 17

6

Find the centre and radius of the following circles:

a
x^2-8x+y^2+4y-5=0
b
x^2+3x+y^2-12y+26=0
Non-monic complete the square
7

Complete the following statements:

a

3 x^{2} + 6 x - 8=⬚\left(x^2 + 2x\right)-8

b

2 x^{2} - 10 x+1=⬚\left(x^2 - 5x\right)+1

c

4 x^{2} +12 x - 6=⬚\left(x^2 + ⬚\right)-6

d

6 x^{2} + 24x - 7=⬚\left(x^2 + ⬚\right)-7

8

Complete the working to rewrite the following in terms of a\left(x + b\right)^2 + c by completing the square:

a
\begin{aligned} 5x^2-10x+4 &= ⬚\left(x^2-2x \right)+4 \\ &=⬚ \left( x^2-2x+⬚ \right)+4-⬚\\ &= ⬚\left(x-⬚ \right)^2-1 \end{aligned}
b
\begin{aligned} 6x^2+36x-1 &= ⬚\left(x^2+⬚ \right)-1 \\ &=⬚ \left( x^2+6x+⬚ \right)-1-⬚\\ &= ⬚\left(x+⬚ \right)^2-⬚ \end{aligned}
9

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

a
2x^2+8x-7
b
5x^2+20x-9
c
4x^2-16x+3
d
10x^2+20x-11
e
7x^2-14x+1
f
9x^2+54x-2
g

3 x^{2} + 33 x + 88

h

5 x^{2} + 5 x + 1

i
4 x^{2} - 11 x + 7
j
3 x^{2} + 9 x + 8
k
3x^2-27x+58
l
2x^2-7x+6
10

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

a
4 x^{2} + 20 x + 16
b
4x^2+20x+28
11

Factorise the following by completing the square to write in the form y = c \left(x + a\right) \left(x + b\right):

a
y = 3 x^{2} + 42 x + 99
b
y=3x^2-12x-36
12

Factorise 2 x^{2} + 11 x + 9 by completing the square to write in the form \left(kx + a\right) \left(x + b\right).

13

A cube has a surface area of 6 x^{2} + 36 x + 54. What is a length of one of the sides?

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Outcomes

MA5.2-8NA

solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques

MA5.3-5NA

selects and applies appropriate algebraic techniques to operate with algebraic expressions

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