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Unit F4 Section 3
Completing the Square

Completing the square is a technique which can be used to solve quadratic equations that do not factorise. It can also be useful when finding the minimum or maximum value of a quadratic.

A general quadratic ax2 + bx + c is written in the form a(x + p)2 + q when completing the square. You need to find the constants p and q so that the two expressions are identical.

Worked Examples

1

Complete the square for x2 + 10x + 2.

First consider the x2 + 10x . These terms can be obtained by expanding (x + 5)2 .

But(x + 5)2 = x2 + 10x + 25
sox2 + 10x = (x + 5)2 − 25
Thereforex2 + 10x + 2 = (x + 5)2 − 25 + 2
= (x + 5)2 − 23
2

Complete the square for x2 + 6x − 8.

To obtain x2 + 6x requires expanding (x + 3)2.

But(x + 3)2 = x2 + 6x + 9
sox2 + 6x = (x + 3)2 − 9
Thereforex2 + 6x − 8 = (x + 3)2 − 9 − 8
= (x + 3)2 − 17

Note

When completing the square for x2 + bx + c, the result is

x2 + bx + c = ⎛
⎝x + ⎞
⎠2 − + c

and for a ≠ 0,

ax2 + bx + c = a⎛
⎝x + ⎞
⎠2 − + c
3

Complete the square for 3x2 + 6x + 7.

As a first step, the quadratic can be rearranged as shown below.

3x2 + 6x + 7 = 3(x2 + 2x) + 7
Then note thatx2 + 2x = (x + 1)2 − 1
so3(x2 + 2x) + 7 = 3⎛
⎝(x + 1)2 − 1⎞
⎠ + 7
= 3(x + 1)2 − 3 + 7
= 3(x + 1)2 + 4
4

(a)

Complete the square for y = 2x2 − 8x + 2.

First rearrange the quadratic as shown.

2x2 − 8x + 2 = 2(x2 − 4x) + 2 .

Then x2 − 4x can be written as (x − 2)2 − 4 to give

2(x2 − 4x) + 2 = 2⎛
⎝(x − 2)2 − 4⎞
⎠ + 2
= 2(x − 2)2 − 8 + 2
= 2(x − 2)2 − 6
(b)

Find the minimum value of y.

As y = 2(x − 2)2 − 6, the minimum possible value of y is − 6, which is obtained when x − 2 = 0 or x = 2.
(c)

Sketch the graph of y = 2x2 − 8x + 2.

Before sketching the graph, it is also useful to find where the curve crosses the x-axis, that is when y = 0. To do this, solve

0 = 2(x − 2)2 − 6
2(x − 2)2 = 6
(x − 2)2 = 3
x − 2 = ±
x = 2±

So the curve crosses the x-axis at 2 + and 2 − , and has a minimum at (2, −6).

This is shown in the graph opposite.
5

(a)

Express 3x2 + 2x + 1 in the form a(x + p)2 + q where a, p and q are real numbers.

As a first step, the quadratic can be rearranged as shown below.

3x2 + 2x + 1 = a(x + p)2 + q
= a(x2 + 2px + p2) + q
= ax2 + 2apx + (ap2 + q)

Equating coefficients:

[x2]3 = a ⇒ a = 3
[x]2 = 2ap ⇒ p = =
[ct]1 = ap2 + q = 3 × ⎛
⎝⎞
⎠2 + q = + q
q = 1 − =

Thus

3x2 + 2x + 1 = 3⎛
⎝x + ⎞
⎠2 +

(b)

Hence, determine for f(x) = 3x2 + 2x + 1

(i)

the minimum value for f(x)

Minimum value of y = 3x2 + 2x + 1 will occur when x + = 0 ; that is, x = , and the value is y = .
(ii)

the equation of the axis of symmetry.

x = is the equation of the axis of symmetry.

Exercises

Complete the square for each of the expressions below.

(a)
x2 + 4x − 5
(b)
x2 + 6x − 1
(c)
x2 + 10x − 2
(d)
x2 − 8x + 2
(e)
x2 + 12x + 3
(f)
x2 − 20x + 10
(g)
x2 + 3x − 1
(h)
x2 − 5x + 2
(i)
x2 − x + 4

Use the completing the square method to solve each of the following equations.

(a)
x2 − 4x + 3 = 0 x = and
(b)
x2 − 6x − 4 = 0 x = and
(c)
x2 + 10x − 8 = 0 x = and
(d)
x2 + 5x + 1 = 0 x = and
(e)
x2 + x − 1 = 0 x = and
(f)
x2 + 2x − 4 = 0 x = and
(g)
x2 + 4x − 8 = 0 x = and
(h)
x2 + 5x − 2 = 0 x = and
(i)
x2 + 7x + 1 = 0 x = and

Complete the square for each of the following expressions.

(a)
2x2 + 8x − 1
(b)
2x2 + 10x − 3
(c)
2x2 + 2x + 1
(d)
3x2 + 6x − 2
(e)
5x2 + 15x − 4
(f)
7x2 − 14x + 2
(g)
3x2 + 12x − 4
(h)
4x2 + 20x − 3
(i)
2x2 − 12x + 3

Sketch the graph of each equation below, showing its minimum or maximum point and where it crosses the x-axis.

Where necessary give answers to 2 decimal places.

(a)

y = x2 − 2x − 1

y = 0 when x = and

the value of y is

(b)

y = x2 + 6x + 8

y = 0 when x = and

the value of y is

(c)

y = x2 − 10x + 24

y = 0 when x = and

the value of y is

(d)

y = x2 + 5x − 14

y = 0 when x = and

the value of y is

(e)

y = 4 + 3x − x2

y = 0 when x = and

the value of y is

(f)

y = 3x − 2 − x2

y = 0 when x = and

the value of y is

The height of a ball thrown into the air is given by

h = 1 + 20t − 10t2

Find the maximum height reached by the ball.

The maximum height of the ball is .
(a)

If 4y2 + 3y + b is a perfect square, calculate the value of b.

(b)

By the method of completing the square, solve the equation 5y2 = 8y − 2.
Give your answers to 3 significant figures.

y = and

Information

The word 'quadratic' comes from the Latin word 'quadratum', which means 'a squared figure'.