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
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