slide 6 - 1 copyright © 2009 pearson education, inc. 6.9 solving quadratic equations by using...
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Slide 6 - 1Copyright © 2009 Pearson Education, Inc.
6.9
Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula
Slide 6 - 2Copyright © 2009 Pearson Education, Inc.
FOIL
A binomial is an expression that contains two terms.
To multiply two binomials, we use the FOIL method.
F = FirstO = OuterI = InnerL = Last
(a + b)(c + d) = ac + ad + bc + bd
F
I
O
L
Slide 6 - 3Copyright © 2009 Pearson Education, Inc.
Example
Multiply: (2x + 4)(x + 6)
644622 xxxx
244122 2 xxx
24162 2 xx
Slide 6 - 4Copyright © 2009 Pearson Education, Inc.
Practice Problem: (x + 7)(x + 2)
Slide 6 - 5Copyright © 2009 Pearson Education, Inc.
To Factor Trinomial Expressions of the Form x2 + bx + c Find two numbers whose product is c and
whose sum is b. Write the factors in the form
(x + ) (x + )
Check your answer by multiplying the factors using the FOIL method.
One number from step 1
Other number from step 1
Slide 6 - 6Copyright © 2009 Pearson Education, Inc.
Factoring Example
Factor x2 7x + 12. We need to find two numbers whose product is
12 and whose sum is 7.
(x 3)(x 4)
3 + 4 = 7
2 + 6 = 8
1 + 12 = 13
Sum of Factors
(3)(4)
(2)(6)
1(12)
Factors of 12
3 + 4 = 7(3)(4)
2 + 6 = 8(2)(6)
1 + 12 = 131(12)
Sum of FactorsFactors of 12
Slide 6 - 7Copyright © 2009 Pearson Education, Inc.
Practice Problem: Factor
122 xx
Slide 6 - 8Copyright © 2009 Pearson Education, Inc.
Factoring Trinomials of the Form ax2 + bc + c, a 1.
Write all pairs of factors of the coefficient of the squared term, a.
Write all pairs of the factors of the constant, c. Try various combinations of these factors until
the sum of the products of the outer and inner terms is bx.
Check your answer by multiplying the factors using the FOIL method.
Slide 6 - 9Copyright © 2009 Pearson Education, Inc.
Example: Factoring
Factor 3x2 + 14x + 8. (3x + )(x + )
Thus, 3x2 + 14x + 8 = (3x + 2)(x + 4).
14x Correct middle term(3x + 2)(x + 4)
10x(3x + 4)(x + 2)
11x(3x + 8)(x + 1)
25x(3x + 1)(x + 8)
Sum of Outer and Inner Terms
Possible Factors
Slide 6 - 10Copyright © 2009 Pearson Education, Inc.
Practice Problem: Factor
1572 2 xx
Slide 6 - 11Copyright © 2009 Pearson Education, Inc.
Solving Quadratic Equations by Factoring
Standard Form of a Quadratic Equation
ax2 + bx + c = 0, a 0
Zero-Factor Property
If a • b = 0, then a = 0 or b = 0.
Slide 6 - 12Copyright © 2009 Pearson Education, Inc.
To Solve a Quadratic Equation by Factoring
Use the addition or subtraction property to make one side of the equation equal to 0.
Factor the side of the equation not equal to 0. Use the zero-factor property to solve the
equation.
Slide 6 - 13Copyright © 2009 Pearson Education, Inc.
Example: Solve by Factoring
Solve 4x2 + 17x 15 = 0.
The solutions are 5 and ¾.
0534 xx
034 x 05x34 x 5x
4
3x
or
Slide 6 - 14Copyright © 2009 Pearson Education, Inc.
Practice Problem: Solve by factoring
07236 2 xx
Slide 6 - 15Copyright © 2009 Pearson Education, Inc.
Quadratic Formula
For a quadratic equation in standard form, ax2 + bx + c = 0, a 0, the quadratic formula is
2 4
2
b b acx
a
Slide 6 - 16Copyright © 2009 Pearson Education, Inc.
Example: Using the Quadratic Formula
Solve the equation 3x2 + 2x 7 = 0.
a = 3, b = 2 and
c = 7
2 4
2
b b acx
a
32
73422 2
6
8442
6
882
6
2222
3
221
Slide 6 - 17Copyright © 2009 Pearson Education, Inc.
Practice Problem: solve using the quadratic equation
0153 2 xx
Slide 6 - 18Copyright © 2009 Pearson Education, Inc.
Homework: P 360 # 7 – 34