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Springboard - Algebra 2
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Find the points on the graph of f(x) where f(x) = 3. The x-coordinates of these points are
the solutions of x2 í 6x + 8 = 3. Because f(x) = 3 when x = 1 and when x = 5, the
solutions of x2 í 6x + 8 = 3 are x = 1 and x = 5.
Lesson 7-2
12. Factor x2 + 11x + 28 by copying and completing the graphic organizer. Then
check by multiplying.
x2 + 11x + 28 = (x + 4)(x + 7)
13. Factor each quadratic expression.
a. 2x2 í 3x í 27
(2x í 9)(x + 3)
b. 4x2 í 121
(2x + 11)(2x í 11)
c. 6x2 + 11x í 10
(2x + 5)(3x í 2)
d. 3x2 + 7x + 4
(3x + 4)(x + 1)
e. 5x2 í 42x í 27
(5x + 3)(x í 9)
f. 4x2 í 4x í 35
(2x + 5)(2x í 7)
g. 36x2 í 100
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4(3x + 5)(3x í 5)
h. 12x2 + 60x + 75
3(2x + 5)2
14. Given that b is positive and c is negative in the quadratic expression x2 + bx +
c, what can you conclude about the signs of the constant terms in the factored
form of the expression? Explain your reasoning.
One constant term is positive, and the other is negative. Sample explanation: The
product of the constant terms is equal to c. If c is negative, the constant terms must
have opposite signs.
15. The area in square inches of a framed photograph is given by the expression
4f2 + 32f + 63, where f is the width in inches of the frame.
a. Factor the quadratic expression.
(2f + 9)(2f + 7)
b. What are the dimensions of the opening in the frame? Explain your answer.
The length is 9 in., and the width is 7 in. The factored expression for the area
shows that the overall length is (2f + 9) in. and the overall width is (2f + 7) in.
The overall length is equal to 2 times f plus the length of the opening, so the
length of the opening is 9 in. The overall width is equal to 2 times f plus the
width of the opening, so the width of the opening is 7 in.
c. If the frame is 2 inches wide, what are the overall dimensions of the framed
photograph? Explain your answer.
13 in. × 11 in.; The expression for the overall length in inches is 2f + 9. If f = 2,
the overall length is 2(2) + 9 = 13 in. The expression for the overall width in
inches is 2f + 7. If f = 2, the overall length is 2(2) + 7 = 11 in.
Lesson 7-3
p. 120
16. Solve each quadratic equation by factoring.
a. 2x2 í 5x í 12 = 0
;x=4
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b. 3x2 + 7x = í 2
; x = í2
c. 4x2 í 20x + 25 = 0
d. 27x2 í 12 = 0
;
e. 6x2 í 4 = 5x
;
17. For each set of solutions, write a quadratic equation in standard form.
a. x = 5, x = í 8
x2 + 3x í 40 = 0
b.
3x2 í 14x + 8 = 0
c.
10x2 + 9x í 7 = 0
d. x = 6
x2 í 12x + 36 = 0
More than one correct equation is possible; other correct equations would be
real-number multiples of the equations given.
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18. A student claims that you can find the solutions of (x í 2)(x í 3) = 2 by
solving the equations x í2 = 2 and x í 3 = 2. Is the student’s reasoning
correct? Explain why or why not.
No. Sample explanation: The student is assuming that if a product is equal to 2,
then one of the factors must be equal to 2. This assumption is incorrect. For
example, the product
is equal to 2, but neither of the factors is equal to 2.
One face of a building is shaped like a right triangle with an area of 2700 ft2.
The height of the triangle is 30 ft greater than its base. Use this information for
Items 19–21.
19. Which equation can be used to determine the base b of the triangle in
feet?
A. b(b + 30) = 2700
B.
C. b(b í 30) = 2700
D.
B
20. Write the quadratic equation in standard form so that the coefficient of b2
is 1.
b2 + 30b í 5400 = 0
21. Solve the quadratic equation by factoring, and interpret the solutions. If
any solutions need to be excluded, explain why.
(b + 90)(b í 60) = 0; b = í90 or b = 60; The solution b = í90 must be excluded,
because b represents the base of a triangle, and it does not make sense for the
base to be negative. The solution b = 60 shows that the base of the triangle
measures 60 ft.
Lesson 7-4
22. For what values of x is the product (x + 4)(x í 6) positive? Explain.
x < í4 or x > 6; Sample explanation: The factor (x + 4) is negative for x < í4
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