Download Section 9.2

Chapter 9
Quadratic
Equations and
Functions
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CHAPTER
9
9.1
9.2
9.3
9.4
Quadratic Equations and
Functions
The Square Root Principle and Completing
the Square
Solving Quadratic Functions Using the
Quadratic Formula
Solving Equations That Are Quadratic in
Form
Graphing Quadratic Functions
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9.2
Solving Quadratic Equations Using the
Quadratic Formula
1. Solve quadratic equations using the quadratic formula.
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Using the Quadratic Formula
To solve a quadratic equation in the form
ax2 + bx + c = 0, where a  0, use the quadratic
formula:
b  b2  4ac
x
2a
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Example
2
2
x
 7x  3  0
Solve.
Solution The equation is in the form ax2 + bx + c = 0,
where a = 2, b = –7, and c = 3.
b  b2  4ac
x
2a
x
  7  
 7   4  2  3
2  2
7  49  24
x
4
2
7  25
x
4
75
x
4
75
x
4
x 3
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75
x
4
1
x
2
5
Example
Solve. x 2  2 x  5  6
Solution First we need to write the equation in the
form ax2 + bx + c = 0. x 2  2 x  11  0 a = 1, b = –2, c = –11.
x
  2  
 2   4 1 11
2  4  44
x
2
2  48
x
2
2
2 1
2  16 3
x
2
24 3
x
2
2 4 3
x 
2
2
x  1 2 3
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Example
Solve. 2 x 2  2  4 x
Solution First we need to write the equation in the
form ax2 + bx + c = 0.
2 x2  4 x  2  0
x
x
4 
 4
2
 4  2  2 
2  2
4  16  4  2  2 
2  2
4  0
x
4
x  1
Notice that the radicand is 0,
which causes this equation to
have only one solution.
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Methods for Solving Quadratic Equations
Method
When the Method is Beneficial
1. Factoring (Section 6.6)
Use when the quadratic equation
can be easily factored.
2. Square root principle
(Section 9.1)
Use when the quadratic equation
can be easily written in the form
3. Completing the square
(Section 9.1)
Rarely the best method, but
important for future topics.
4. Quadratic formula
(Section 9.2)
Use when factoring is not easy, or
possible, with integer coefficients.
ax 2  c, or (ax  b)2  c.
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