Chapter Chapter 9 9 Section Section 3 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Chapter 9 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Jan 02, 2016
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Chapter Chapter 99Section Section 33
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Quadratic Equations by the Quadratic Formula
Identify the values of a, b, and c in a quadratic equation.Use the quadratic formula to solve quadratic equations.Solve quadratic equations with only one solution.Solve quadratic equations with fractions.
11
44
33
22
9.39.39.39.3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Quadratic Equations by the Quadratic Formula
We can solve any quadratic equation by completing the square, but the method is tedious. In this section, we complete the square on the general quadratic equation
ax2 + bx + c = 0 (where a does not equal 0).
Slide 9.3 - 3
By doing this, we get the quadratic formula, which gives the solutions of any quadratic equation.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Identify the values of a, b, and c in a quadratic equation.
Slide 9.3 - 4
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Identify the values of a, b, and c in a quadratic equation.
To solve a quadratic equation with the quadratic formula, we must first identify the values of a, b, and c in the standard form of the quadratic equation.
Slide 9.3 - 5
The quadratic formula should be memorized and the values of a, b, and c are only determined after the equation is written in standard form.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1
Write the equation in standard form, if necessary, and then identify the values of a, b, and c.
Solution:
Determining Values of a, b, and c in Quadratic Equations
Slide 9.3 - 6
25 2 1 0x x 23 2x x
29 13 0x 3 2 1 7m m
23 1 2 7m 23 3 6 7m
23 3 13 0m
5, 2, 1a b c
9, 0, 13a b c
3, 3, 13a b c
23 2 0x x 3, 1, 2a b c
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Use the quadratic formula to solve quadratic equations.
Slide 9.3 - 7
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
To develop the quadratic formula, we follow the steps given in Section 9.2 for completing the square on ax2 + bx + c = 0. This formula is also valid, however, for a < 0. Notice that there are two values: one for the + sign and one for the – sign.
Use the quadratic formula to solve quadratic equations.
Slide 9.3 - 8
Notice that the fraction bar is under –b as well as the radical. When using this formula, be sure to find the values of first. Then divide those results by the value of 2a.
2 4b b ac
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve 2x2 + x – 3 = 0.
EXAMPLE 2
Solution:
Solving a Quadratic Equation by the Quadratic Formula
Slide 9.3 - 9
2, 1, 3a b c
2 4
2
b b acx
a
21 1 4 2 3
2 2x
1 25
4x
1 1 24
4x
1 5
4x
1 5
4x
or
4
4x
6 3
4 2x
or
3,1
2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3
Solution:
Rewriting a Quadratic Equation before Solving
Slide 9.3 - 10
Solve –x2 = 8x + 1.
1, 8, 1a b c
2 4
2
b b acx
a
28 8 4 1 1
2 1x
8 60
2x
8 64 4
2x
8 2 15
2x
8 2 15
2x
or
4 15x 4 15x or
4 15
2 8 1 0x x 8 4 15
2x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 33
Slide 9.3 - 11
Solve quadratic equations with only one solution.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve quadratic equations with only one solution.
In the quadratic formula, the quantity under the radical, b2 – 4ac, is called the discriminant. When the discriminant equals 0, the equation has just one rational number solution, and the trinomial ax2 + bx + c is a perfect square.
Slide 9.3 - 12
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Solution:
Solving a Quadratic Equation with Only One Solution
Slide 9.3 - 13
Solve 9x2 = 42x – 49.
9, 42, 49a b c
242 42 4 9 49
2 9x
42 1764 1764
18x
42
18x
42 0
18x
7
3
29 42 49 0x x
7
3x
2 4
2
b b acx
a
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 44
Slide 9.3 - 14
Solve quadratic equations with fractions.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5A
Solve.
Solution:
2 2 4
3 9m m
Solving a Quadratic Equation with Fractions
Slide 9.3 - 15
2 6 49 9
9 9m m
29 6 4m m
29 6 4 0m m
9, 6, 4a b c 2 4
2
b b acm
a
26 6 4 9 4
2 9m
6 36 144
18m
1 5
3m
1 5
3m
or
1 5
3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5B
Solve.
Solution:
2 4 2
3 3x x
Solving a Quadratic Equation with Fractions
Slide 9.3 - 16
2 4 23 3
3 3x x
23 4 2 0x x
3, 4, 2a b c
2 4
2
b b acx
a
24 4 4 3 2
2 3x
4 16 24
6x
4 8
6x
Because of the negative square root there is no solution, so the solution set is Ø.
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