Solve quadratic equations by completing the square.

1
Objectives
Solve quadratic equations by completing the
square. Write quadratic equations in vertex
form.
2
Notes
1. Complete the square for the expression x2
15x . Write the resulting expression as a
binomial squared.
Solve each equation.
3. x2 4x1
2. x2 16x 64 20
Write each function in vertex form and identify
its vertex.
5. f(x) x2 12x 27
4. f(x) x2 6x 7
3
Example 1A Solving Equations by Using the Square
Root Property
Solve the equation.
4x2 11 59
Subtract 11 from both sides.
4x2 48
Divide both sides by 4 to isolate the square
term.
x2 12
Take the square root of both sides.
Simplify.
4
Example 1B
Solve the equation.
x2 8x 16 49
Factor the perfect square trinomial.
(x 4)2 49
Take the square root of both sides.
Subtract 4 from both sides.
x 11, 3
Simplify.
5
Example 1C
Solve the equation.
x2 12x 36 28
Factor the perfect square trinomial
(x 6)2 28
Take the square root of both sides.
Subtract 6 from both sides.
Simplify.
6
If a quadratic expression of the form x2 bx
cannot model a square, you can add a term to form
a perfect square trinomial. This is called
completing the square.
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Example 2A Solving a Quadratic Equation by
Completing the Square
Solve the equation by completing the square.
x2 12x 20
Collect variable terms on one side.
x2 12x 20
Set up to complete the square.
Simplify.
x2 12x 36 20 36
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Example 2A Continued
Factor.
(x 6)2 16
Take the square root of both sides.
Simplify.
x 6 4
x 6 4 or x 6 4
Solve for x.
x 10 or x 2
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Example 2B
Solve the equation by completing the square.
3x2 24x 27
Divide both sides by 3.
x2 8x 9
Set up to complete the square.
Simplify.
10
Example 2B Continued
Solve the equation by completing the square.
Factor.
Take the square root of both sides.
Solve for x.
x 4 5 or x 4 5
x 1 or x 9
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Example 2B imaginary version
Solve the equation by completing the square.
Factor.
(x 4)2 -25
Take the square root of both sides.
x 4 5i or x 4 -5i
Solve for x.
x 4 5i or 4 5i
12
Recall the vertex form of a quadratic function
from lesson 5-1 f(x) a(x h)2 k, where the
vertex is (h, k).
You can complete the square to rewrite any
quadratic function in vertex form.
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Example 3A Writing a Quadratic Function in
Vertex Form
Write the function in vertex form, and identify
its vertex.
f(x) x2 16x 12
Set up to complete the square.
Simplify and factor.
f(x) (x 8)2 76
Because h 8 and k 76, the vertex is (8,
76).
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Example 3A Continued
Check Use the axis of symmetry formula to
confirm vertex.
?
y f(8) (8)2 16(8) 12 76
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Example 3B
Write the function in vertex form, and identify
its vertex
f(x) x2 24x 145
Set up to complete the square.
Simplify and factor.
f(x) (x 12)2 1
Because h 12 and k 1, the vertex is (12,
1).
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Example 3B Continued
Check Use the axis of symmetry formula to
confirm vertex.
y f(12) (12)2 24(12) 145 1
?
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Notes
1. Complete the square for the expression x2
15x . Write the resulting expression as a
binomial squared.
Solve each equation.
3. x2 4x1
2. x2 16x 64 20
Write each function in vertex form and identify
its vertex.
5. f(x) x2 12x 27
4. f(x) x2 6x 7
f(x) (x 6)2 36 (6, 36)
f(x) (x 3)2 16 (3, 16)
18
You can complete the square to solve quadratic
equations.