Completing the squareSolving quadratic equations

1.

Express the followings in completed square form and hence solve the equations

x2 + 4x – 12 = 0

(x + 2)2 – 16 = 0

(x + 2)2 = 16

x + 2 = 16 x + 2 = 4 x = - 2 4 x = -6 or x = 2

= (x + 2)2 – 22 – 12 = 0

2. x2 + 6x + 4 = 0

(x + 2)2 – 5 = 0

(x + 3)2 = 5

x + 3 = 5 x = - 3 5 x = - 3 - 5 or - 3 + 5

= (x + 3)2 – 32 + 4 = 0

Sketching graphExpress x2 - 4x -5 in the form (x + p)2 + q, hence:

i) find the minimum value of the expression y = x2 - 4x - 5 .

ii) solve the equation x2 - 4x - 5 = 0

iii) sketch the graph of the function y = x2 - 4x - 5

Completed square form

x2 – 4x – 5 =

y

x

Vertex (2, -9) The curve is symmetrical about x = 2

(x – 2)2 – 9

x2 – 4x – 5 = (x – 2)2 – 9 = 0

x – 2 = 9x – 2 = 3x = 2 3x = -1 or x = 5

(-1, 0) (5, 0)

(x – 2)2 – 4 - 5 =

Solving: x2 – 4x – 5 = 0

(x – 2)2 = 9

Sketching graph

Write 1 + 4x - x2 in completed square form, hence solve 1 + 4x – x2 = 0 and sketch the graph of y = 1 + 4x – x2.

Completed square form

1 + 4x – x2 = - [ x2 – 4x ] + 1

y

x

Vertex (2, 5)

The curve is symmetrical about x = 2

-[ x2 – 4x ] + 1 = - [ (x – 2)2 – 4 ] + 1 = - (x – 2)2 + 4 + 1

= - (x – 2)2 + 5 - (x – 2)2 + 5 = 0

- (x – 2)2 = - 5

(x – 2)2 = 5

x – 2 = 5

x = 2 5

x = 2 -5 or x = 2 + 5

(2 - 5)(2 + 5)

Sketching graphWrite -3x2 + 6x - 2 in completed square form, hence solve -3x2 + 6x – 2 and sketch the graph of y = -3x2 + 6x – 2.

Completed square form

-3[ x2 - 2x ] – 2 = -3[ (x - 1)2 - 1 ] - 2

y

x

Vertex ( 1, 1 )

The curve is symmetrical about x = 1

= -3(x - 1)2 + 3 - 2

= -3(x - 1)2 + 1

-3(x - 1)2 + 1 = 0

-3(x - 1)2 = - 1

31)1( 2 x

311 x

311x

(1 -(1/3), 0)(1 +(1/3), 0)

More examplesComplete the square for each of the following quadratic functions and solve f(x) = 0

(a) x2 + x – ½ = (x + ½ )2 – ¼ – ½ = (x + ½ )2 – ¾

(c) 3 + 4x – 2x2 = -2 [x2 + 2 x ] + 3 = 2[(x + 1 )2 – 1 ] + 3

= 2(x + 1 )2 – 2 + 3 = 2(x + 1 )2 + 1

(x + ½ )2 – ¾ = 0

(x + ½ )2 = ¾

x + ½ = ¾

x = -½ ¾

2(x + 1 )2 + 1 = 0

2(x + 1 )2 = - 1 (x + 1 )2 = - ½ No solution

The function f is defined for all x by f(x) = x2 + 3x – 5.

a) Express f(x) in the form (x + P)2 + Q.

Complete the square

23 9( ) 5

2 4x 2 3 5x x

23 9 20( )

2 4 4x

23 29( )

2 4x

Solve the equation f(x) = 0 by making x the subject, using the completed square format

b) Hence, or otherwise, solve the equation f(x) = 0, giving your answers in surd form.

23 29( ) 0

2 4x

23 29 ( )

2 4

29( )

4x

3 29 (Square root)

2 4x

3 29 )

2

3

4(

2x

3 29

2 2x

Tip: You could have used the quadratic formula on x2 + 3x – 5 = 0.

Tip: Simplify the surd where aa

b b