Name———————————————————————— Lesson
1.4
Date —————————————
Study Guide
For use with the lesson “Solve ax 2 + bx + c = 0 by Factoring”
example 1
Use factoring to solve equations of the form ax 2 1 bx 1 c 5 0.
Lesson 1.4
goal
Factor ax 2 1 bx 1 c where c < 0
Factor 2x 2 2 x 2 3.
You want 2x 2 2 x 2 3 5 (kx 1 m)(lx 1 n) where k and l are factors of 2 and m and
n are factors of 23. Because mn < 0, m and n have opposite signs.
k, l
2, 1
2, 1
2, 1
2, 1
m, n
3, 21
21, 3
23, 1
1, 23
(kx 1 m)(lx 1 n) (2x 1 3)(x 2 1) (2x 2 1)(x 1 3) (2x 2 3)(x 1 1) (2x 1 1)(x 2 3)
ax 2 1 bx 1 c
2x 2 1 x 2 3
2x 2 1 5x 2 3
2x 2 2 x 2 3
2x 2 2 5x 2 3
The correct factorization is (2x 2 3)(x 1 1).
example 2
Factor with special patterns and monomials
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Factor the expression.
a. 6t 2 2 24 5 6(t 2 2 4)
5 6(t 1 2)(t 2 2)
2
b. 3m 2 18m 1 27 5 3(m 22 6m 1 9)
5 3(m 2 3)2
example 3
Factor out monomial.
Difference of two squares
Factor out monomial.
Perfect square trinomial
Solve a quadratic equation
4s 2 1 11s 1 8 5 3s 1 4
Original equation
4s 2 1 8s 1 4 5 0
Write in standard form.
s 2 1 2s 1 1 5 0
Divide each side by 4.
(s 1 1)2 5 0
s 1 1 5 0
s 5 21
Factor.
Zero product property
Solve for s.
Exercises for Examples 1, 2, and 3
Factor the expression. If the expression cannot be factored, say so.
1. 2x 2 2 14x 1 12
2. 3x 2 1 24x 1 21
3. 6x 2 2 42x 1 72
4. 4x 2 1 4x 2 24
5. 5x 2 2 20x 2 25
6. 3x 2 2 12x 2 36
8. 3x 2 2 12x 1 12 5 0
9. 5x 2 2 15x 2 20 5 0
Solve the equation.
7. 25x 2 2 9 5 0
Algebra 2
Chapter Resource Book
1-49
Name———————————————————————— Lesson
Lesson 1.4
1.4
Date —————————————
Study Guide continued
For use with the lesson “Solve ax 2 + bx + c = 0 by Factoring”
example 4
Use a quadratic equation as a model
Paintings The area of a painting is 24 square inches and the length is 5 inches more
than the width. Find the length of the painting.
Solution
Write a verbal model. Let x represent the width and write an equation.
Area of painting 5 (square inches)
24
Width of painting p (inches)
x
5
Length of painting
(inches)
p
(x 1 5)
0 5 x 2 1 5x 2 24
Write in standard form.
0 5 (x 1 8)(x 2 3)
Factor.
x 1 8 5 0
or
x 2 3 5 0
Zero product property
x 5 28
or
x 5 3
Solve for x.
Reject the negative value, x 5 28. The length is x 5 3 1 5 inches or 8 inches.
example 5
Solve a multi-step problem
STEP 1 Define the variables. Let x represent the price increase and R(x) represent the
weekly revenue.
STEP 2 Write a verbal model. Then write and simplify a quadratic equation.
Weekly sales 5 (dollars)
R(x)
5
Number of bikes p sold (bikes)
Price of bike
(dollars/bike)
p
(100 1 10x)
(18 2 3x)
R(x) 5 230(x 2 6)(x 1 10)
STEP 3 Identify the zeros and find their average. The zeros are 6 and 210. The
average of the zeros is 22. To maximize revenue, the shop should charge
100 1 10(22) 5 $80.
STEP 4 Find the maximum weekly revenue.
R(22) 5 230(22 2 6)(22 1 10) 5 $1920.
The shop should charge $80 per bike to maximize weekly revenue.
The maximum weekly revenue is $1920.
Exercises for Examples 4 and 5
10. Rework Example 4 where the length is 2 inches more than the width.
11. Rework Example 5 where for each increase of $10, the shop sells 9 less bikes
per week.
1-50
Algebra 2
Chapter Resource Book
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Bicycles A bicycle shop sells about 18 bikes per week when it charges $100 per bike.
For each increase of $10, the shop sells 3 less bikes per week. How much should the
shop charge to maximize sales?
24; they are the same.
19x; they are the same.
6. The trinomial and product of binomials in
Question 1 are different. The trinomial and
product of binomials in Question 5 are equivalent.
Practice Level A
1. (2x 1 1)(x 1 2) 2. (2x 1 1)(x 1 1)
3. (3x 2 1)(x 1 2) 4. (2x 2 1)(x 2 1)
5. 2(2x 2 1)(x 1 1) 6. (3x 1 1)(2x 2 3)
7. (5x 2 4)(x 1 1) 8. 3(3x 1 2)(x 2 1)
9. (4x 1 1)(x 1 3) 10. 2(3x 2 2)(x 1 1)
11. (2x 1 3)(2x 2 3) 12. 4(x 2 1)(x 1 1)
13. 2(x 2 2)(x 2 3) 14. 3(x 1 1)(x 2 4)
15. 3(3x 2 1)(3x 1 1) 16. 4(2x 1 1)(x 2 3)
17. 2(2x 1 3)(2x 1 3) 18. 5(3x 1 2)(2x 2 1)
3
1
1
1 2
19. }
​ 2 ​, 1 20. 2​ }2 ​, 21 21. }
​ 2 ​, }
​ 3 ​ 22. 2​ }3 ​, 3
3 5
1
3
23. ​ } ​, 1 24. 2​ } ​, ​ } ​ 25. 22, 3 26. 2​ } ​, 3
2 2
2
4
2
27. 21, }
​ 3 ​ 28. 1, 2 29. 23, 1 30. 24, 2
5
3
1 1
3
31. ​ } ​ 32. 2​ } ​, ​ } ​ 33. 2​ } ​, 2​ } ​
3 3
2
2
2
34. 3 35. 2 36. B 37. 1.5 ft
Practice Level B
1. (3x 2 2)(x 1 4) 2. (2x 2 1)(x 1 3)
Practice Level C
1. (4x 2 3)(x 2 3) 2. (3x 1 4)(2x 2 7)
3. (5x 1 2)(2x 1 5) 4. (3x 1 4)(3x 1 4)
5. 23(2x 2 3)(2x 1 3) 6. 4(3x 1 2)(x 2 4)
7. 23(5x 1 4)(x 2 1) 8. x(3x 2 1)(2x 2 1)
9. cannot factor 10. x 2(2x 1 3)(2x 1 3)
11. 3x 2(4x 2 3)(2x 1 3) 12. 4x(6x 2 5)(3x 2 7)
13. (x 2 1 9)(x 2 3)(x 1 3)
14. (2x 2 1 3)(x 2 1 1)
15. 3(2x 2 2 1)(x 2 1)(x 1 1)
3 3
5
1
1
16. }
​ 3 ​, 3 17. 1, }
​ 2 ​ 18. 2​ }4 ​, }
​ 4 ​ 19. 21, }
​ 6 ​
3 7
3 5
4 3
4 7
20. 2​ } ​, ​ } ​ 21. 2​ } ​, ​ } ​ 22. 2​ } ​, }
​   ​ 23. 2​ }4 ​, }
​ 7 ​
3 2
2 9
3 4
2
2
3
2 7
24. 2​ } ​, 4 25. 1, 2 26. 2​ } ​, 2 27. ​ } ​, }
​   ​ 28. }
​ 2 ​
5
3
3 5
29. 10 30. 99x 2 1 2900x 2 38,900 5 0; 100 ft
Study Guide
1. 2(x 2 6)(x 2 1) 2. 3(x 1 1)(x 1 7)
3. 6(x 2 3)(x 2 4) 4. 4(x 2 2)(x 1 3)
5. 5(x 1 1)(x 2 5) 6. 3(x 2 6)(x 1 2)
3
7. x 56​ } ​  8. x 5 2 9. x 5 21, x 5 4
5
10. The length is 6 in. 11. The shop should
charge $60/bike to maximize weekly revenue of
$3240.
Interdisciplinary Application
3. (2x 1 1)(2x 1 1) 4. cannot factor
1. about 210 yd 2. about 60 yd
5. (4x 2 3)(x 1 2) 6. (2x 1 5)(x 1 3)
3. about 287 yd 4. about 361 yd
7. (3x 1 2)(3x 1 2) 8. 3(2x 2 3)(2x 2 1)
Challenge Practice
9. 2(3x 2 1)(3x 1 1) 10. (4x 1 3)(3x 1 2)
1. Sample answer: c 5 23, (2x 2 1)(x 1 3);
c 5 212, (2x 2 3)(x 1 4)
2. Sample answer: c 5 28, (3x 1 2)(x 2 4);
c 5 225, (3x 1 5)(x 2 5)
11. (5x 2 4)(3x 1 4) 12. cannot factor
13. 3(4x 2 1)(x 2 3) 14. (6x 2 7)(3x 1 2)
15. 2(5x 2 6)(2x 2 3) 16. 7(3x 1 1)(2x 1 1)
17. (212x 1 11)(x 1 1) 18. 4(5x 1 3)(4x 1 1)
3
2 2
1
1 3
19. 22, }
​ 2 ​ 20. 2​ }2 ​, 3 21. }
​ 2 ​, }
​ 2 ​ 22. 2​ }3 ​, }
​ 3 ​
5
4
2
1 3
1 1
23. }
​ 4 ​, }
​ 2 ​ 24. 2​ }3 ​ 25. 2​ }3 ​, 2​ }3 ​ 26. 2​ }2 ​, }
​ 2 ​
3 5
2
3
3 3
27. 0, }
​ 7 ​ 28. }
​ 8 ​ , }
​ 4 ​  29. 2​ }3 ​, 0 30. 2​ }2 ​, }
​ 6 ​
6 4
8
31. 2, 3 32. 2​ } ​, }
​   ​ 33. 2​ }
  ​,  1 34. none
5 5
11
A8
3
35. 2 36. }
​ 2 ​ 37. 0.375 ft
Algebra 2
Chapter Resource Book
1 
2
4 11
3. (22, 23), ​ }
​ 3 ​, }
​ 3  ​  ​
3
4. ​ 2}
​ 2 ​, 24  ​, (3, 23)
1 
2
5. (2n 2 2)(2n)(2n 1 2); An even integer is one
that is a multiple of 2. Because n is an integer,
2n is an even integer. You can add and subtract 2
from 2n to obtain a set of three consecutive even
integers. 6. 5 7. 24 club members
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Lesson 1.4 Solve ax 2 + bx + c
= 0 by Factoring, continued