Unit 17 Section 3 : Quadratic Equations: Completing the Square

Completing the square is a useful technique for solving quadratic equations. It is a more powerful technique than factorisation because it can be applied to equations that do not factorise.

When completing the square, an expression like,

ax^2 + bx + c is written in the form (Ax + B)^2 + C.

We will begin with the simple example where a = 1. In this case we will write expressions in the form

x^2 + bx + c as (x + B)^2 + C

If we expand (x + B)^2 + C we get x^2 + 2Bx + B^2 + C.

Comparing this with x^2 + bx + c shows that

b = 2B and c = B^2 + C

which gives B = and C = c - B^2

Using these two results we can now set about completing the square in some simple cases.

Example 1

Write each of the following expressions in the form (x + B)^2 + C.

(a)

x^2 + 6x + 1

Comparing x^2 + 6x + 1 with x^2 + bx + c we see that b = 6 and c = 1 in this case, so

B = = = 3   and   C = c - B^2 = 1 - 3^2 = -8.

Therefore x^2 + 6x + 1 = (x + 3)^2 - 8.

(b)

x^2 + 4x - 2

Here b = 4 and c = –2, so

B = = = 2   and   C = c - B^2 = (-2) - 2^2 = -6.

Therefore x^2 + 4x - 2 = (x + 2)^2 - 6.

(c)

x^2 + 2x

Here B = = 1 and C = 0 - 1^2 = -1,
so   x^2 + 2x = (x + 1)^2 - 1.

Example 2

Solve the following equations by completing the square.

(a)

x^2 - 4x - 5 = 0

Completing the square gives,

x^2 - 4x - 5 = (x - 2)^2 - 9

Now we can solve the equation

(x – 2)2 = 9
x – 2 = ±
x – 2 = ± 3
x = 2 ± 3
sox = 5   or   –1
(b)

x^2 + 6x - 1 = 0

Completing the square gives,

x^2 + 6x - 1 = (x + 3)^2 - 10

Now we can solve the equation

(x + 3)2 = 10
x + 3 = ±
x = –3 ±
sox = 0.162   or   –6.162   to 3 decimal places

Exercises

Note: To type indeces on this page use ^ sign. e.g.   n2:  
Question 1

Write each of the following expressions in the form (x + B)2 + C.

(a)x^2 + 6x ( )2
(b)x^2 + 4x ( )2
(c)x^2 + 8x ( )2
(d)x^2 - 10x ( )2
(e)x^2 + 7x ( )2
(f)x^2 - 5x ( )2
Question 2

Write each of the following expressions in the form (x + B)2 + C:

(a)x^2 + 6x + 1 ( )2
(b)x^2 - 8x + 3 ( )2
(c)x^2 + 10x - 12 ( )2
(d)x^2 + 12x + 8 ( )2
(e)x^2 - 4x + 1 ( )2
(f)x^2 - 6x - 3 ( )2
(g)x^2 + 5x + 3 ( )2
(h)x^2 + 3x - 4 ( )2
(i)x^2 + x - 2 ( )2
(j)x^2 - x + 3 ( )2
Question 3

Solve each of the following quadratic equation by completing the square:

(a)x^2 - 2x - 8 = 0 x =   or   x =
(b)x^2 + 4x + 3 = 0 x =   or   x =
(c)x^2 + 8x + 12 = 0 x =   or   x =
(d)x^2 - 5x + 4 = 0 x =   or   x =
(e)x^2 - 2x - 15 = 0 x =   or   x =
(f)x^2 + 3x - 28 = 0 x =   or   x =
Question 4

Solve each of the following quadratic equations by completing the square.
Give your answers to 2 decimal places.

(a)x^2 + 2x - 5 = 0 x =   or   x =
(b)x^2 + 4x - 1 = 0 x =   or   x =
(c)x^2 + 6x - 5 = 0 x =   or   x =
(d)x^2 - 10x - 1 = 0 x =   or   x =
(e)x^2 + x - 3 = 0 x =   or   x =
(f)x^2 - 3x + 1 = 0 x =   or   x =
(g)x^2 + 5x - 4 = 0 x =   or   x =
(h)x^2 + 3x - 5 = 0 x =   or   x =
Question 5

The rectangle shown has an area of 20 cm² .

(a)

Write down an equation for the width x of the rectangle.

This equation can be simplified to x^2 + 4x - 20 = 0.
(b)

Use completing the square to determine the width of the rectangle to 2 decimal places.

Width = cm
Question 6

(a)

Write the equation x^2 - 8x + 18 in the form (x + B)2 + C = 0.

( )2
(b)

How many solutions has this equation?

The equation has no solution, because (x - 4)^2 cannot be negative.
Question 7

Simplify each of the following equations and obtain their solutions by completing the square.

(a)4x^2 + 20x - 8 = 0 x =   or   x =
(b)20x^2 - 40x + 60 = 0 x =   or   x =
(c)3x^2 + 6x - 9 = 0 x =   or   x =
(d)5x^2 - 30x - 15 = 0 x =   or   x =
Question 8

The height of a ball at time t seconds can be calculated by using the formula

h = 20t - 5t^2

(a)

Calculate the value of h when t = 2.

h =
(b)

Determine the values of t for which h = 15.

t =   or   t =
Question 9

The area of the rectangle shown is 30 cm².
Determine the value of x.

x =

Question 10

The area of the triangle shown is 120 cm².
Determine the perimeter of the triangle correct to the nearest millimetre.

Perimeter = cm
Question 11

Use the method of completing the square or the appropriate formula to solve x^2 + 4x - 2 = 0.
Write your answers to 2 decimal places.

x =   or   x =