10.7
Before
Now
Why?
Key Vocabulary
• discriminant
Interpret the
Discriminant
You used the quadratic formula.
You will use the value of the discriminant.
So you can solve a problem about gymnastics, as in Ex. 49.
In the quadratic formula, the expression b2 2 4ac is called the discriminant
of the associated equation ax2 1 bx 1 c 5 0.
}
6 Ï b2 2 4ac
x 5 2b
}}
2a
discriminant
Because the discriminant is under the radical symbol, the value of the
discriminant can be used to determine the number of solutions of a
quadratic equation and the number of x-intercepts of the graph of the
related function.
For Your Notebook
KEY CONCEPT
Using the Discriminant of ax 2 1 bx 1 c 5 0
READING
Recall that in this
course, solutions
refers to real-number
solutions.
Value of the
discriminant
b2 2 4ac > 0
b2 2 4ac 5 0
b2 2 4ac < 0
Number of
solutions
Two solutions
One solution
No solution
Graph of
y 5 ax2 1 bx 1 c
y
y
x
x
Two x-intercepts
EXAMPLE 1
One x-intercept
x
No x-intercept
Use the discriminant
Equation
ax2 + bx + c = 0
Discriminant
b2 – 4ac
Number of
solutions
a. 2x2 1 6x 1 5 5 0
62 2 4(2)(5) 5 24
No solution
b. x2 2 7 5 0
02 2 4(1)(27) 5 28
Two solutions
2
c. 4x 2 12x 1 9 5 0
678
y
Chapter 10 Quadratic Equations and Functions
2
(212) 2 4(4)(9) 5 0
One solution
EXAMPLE 2
Find the number of solutions
Tell whether the equation 3x 2 2 7 5 2x has two solutions, one solution, or
no solution.
Solution
STEP 1 Write the equation in standard form.
3x2 2 7 5 2x
2
3x 2 2x 2 7 5 0
Write equation.
Subtract 2x from each side.
STEP 2 Find the value of the discriminant.
b2 2 4ac 5 (22)2 2 4(3)(27)
5 88
Substitute 3 for a, 22 for b, and 27 for c.
Simplify.
c The discriminant is positive, so the equation has two solutions.
✓
GUIDED PRACTICE
for Examples 1 and 2
Tell whether the equation has two solutions, one solution, or no solution.
1. x2 1 4x 1 3 5 0
EXAMPLE 3
2. 2x2 2 5x 1 6 5 0
3. 2x2 1 2x 5 1
Find the number of x-intercepts
Find the number of x-intercepts of the graph of y 5 x 2 1 5x 1 8.
Solution
Find the number of solutions of the equation 0 5 x2 1 5x 1 8.
b2 2 4ac 5 (5)2 2 4(1)(8)
5 27
Substitute 1 for a, 5 for b, and 8 for c.
Simplify.
c The discriminant is negative, so the equation has no solution. This means
that the graph of y 5 x2 1 5x 1 8 has no x-intercepts.
CHECK
✓
You can use a graphing calculator to
check the answer. Notice that the graph
of y 5 x2 1 5x 1 8 has no x-intercepts.
GUIDED PRACTICE
for Example 3
Find the number of x-intercepts of the graph of the function.
4. y 5 x2 1 10x 1 25
5. y 5 x2 2 9x
6. y 5 2x2 1 2x 2 4
10.7 Interpret the Discriminant
679
EXAMPLE 4
Solve a multi-step problem
FOUNTAINS The Centennial Fountain in Chicago shoots a water arc that can
be modeled by the graph of the equation y 5 20.006x2 1 1.2x 1 10 where x is
the horizontal distance (in feet) from the river’s north shore and y is the height
(in feet) above the river. Does the water arc reach a height of 50 feet? If so,
about how far from the north shore is the water arc 50 feet above the water?
y
50
North shore
x
50
Solution
STEP 1 Write a quadratic equation. You want to know whether the water arc
reaches a height of 50 feet, so let y 5 50. Then write the quadratic
equation in standard form.
y 5 20.006x2 1 1.2x 1 10
2
50 5 20.006x 1 1.2x 1 10
0 5 20.006x2 1 1.2x 2 40
Write given equation.
Substitute 50 for y.
Subtract 50 from each side.
STEP 2 Find the value of the discriminant of 0 5 20.006x2 1 1.2x 2 40.
b2 2 4ac 5 (1.2)2 2 4(20.006)(240)
5 0.48
a 5 20.006, b 5 1.2, c 5 240
Simplify.
STEP 3 Interpret the discriminant. Because the discriminant is positive,
the equation has two solutions. So, the water arc reaches a height of
50 feet at two points on the water arc.
STEP 4 Solve the equation 0 5 20.006x2 1 1.2x 2 40 to find the distance
from the north shore where the water arc is 50 feet above the water.
}
6 Ïb 2 4ac
x 5 2b
}}
2
2a
Quadratic formula
}
USE A SHORTCUT
Because the value of
b2 – 4ac was calculated
in Step 2, you can
substitute 0.48 for
b2 – 4ac.
✓
21.2 6 Ï0.48
5}
2(20.006)
x ø 42 or x ø 158
Substitute values in the quadratic formula.
Use a calculator.
c The water arc is 50 feet above the water about 42 feet from the north shore
and about 158 feet from the north shore.
GUIDED PRACTICE
for Example 4
7. WHAT IF? In Example 4, does the water arc reach a height of 70 feet?
If so, about how far from the north shore is the water arc 70 feet above
the water?
680
Chapter 10 Quadratic Equations and Functions
10.7
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 9 and 47
★ 5 STANDARDIZED TEST PRACTICE
Exs. 2, 18, 19, 40, 41, and 47
SKILL PRACTICE
1. VOCABULARY Write the quadratic formula and circle the expression that
represents the discriminant.
2.
★
WRITING Explain how the discriminant of ax2 1 bx 1 c 5 0 is related
to the graph of y 5 ax2 1 bx 1 c.
EXAMPLES
1 and 2
on pp. 678–679
for Exs. 3–21
USING THE DISCRIMINANT Tell whether the equation has two solutions, one
solution, or no solution.
3. x2 1 x 1 1 5 0
4. 2x2 2 5x 2 6 5 0
5. 22x2 1 8x 2 4 5 0
6. 3m2 2 6m 1 7 5 0
7. 9v 2 2 6v 1 1 5 0
8. 23q2 1 8 5 0
9. 25p2 2 16p 5 0
10. 2h2 1 3 5 4h
11. 10 5 x2 2 5x
1 2
12. }
z 125z
4
13. 23g 2 2 4g 5 }
14. 8r 2 1 10r 2 1 5 4r
15. 3n2 1 3 5 10n 2 3n2
16. 8x2 1 9 5 4x2 2 4x 1 8
17. w 2 2 7w 1 29 5 4 2 7w
3
4
18.
★
MULTIPLE CHOICE What is the value of the discriminant of the
equation 5x2 2 7x 2 2 5 0?
A 29
19.
★
B 9
C 59
D 89
MULTIPLE CHOICE How many solutions does 2x2 1 4x 5 8 have?
A None
B One
C Two
D Three
ERROR ANALYSIS Describe and correct the error in finding the number of
solutions of the equation.
20. 4x2 1 12x 1 9 5 0
21. 3x2 2 7x 2 4 5 29
b2 2 4ac 5 122 2 4(4)(9)
b2 2 4ac 5 (27) 2 2 4(3)(24)
5 144 2 144
5 49 2 (248)
50
5 97
The equation has two solutions.
The equation has two solutions.
EXAMPLE 3
FINDING THE NUMBER OF x-INTERCEPTS Find the number of x-intercepts of
on p. 679
for Exs. 22–30
the graph of the function.
22. y 5 x2 2 2x 2 4
23. y 5 2x2 2 x 2 1
24. y 5 4x2 1 4x 1 1
25. y 5 2x2 2 5x 1 5
26. y 5 x2 2 6x 1 9
27. y 5 6x2 1 x 1 2
28. y 5 213x2 1 2x 1 6
1 2
29. y 5 }
x 2 3x 1 9
2 2
30. y 5 }
x 2 5x 1 12
4
3
REASONING Give a value of c for which the equation has (a) two solutions,
(b) one solution, and (c) no solution.
31. x2 2 2x 1 c 5 0
32. x2 2 8x 1 c 5 0
33. 4x2 1 12x 1 c 5 0
10.7 Interpret the Discriminant
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USING THE DISCRIMINANT Tell whether the vertex of the graph of the
function lies above, below, or on the x-axis. Explain your reasoning.
34. y 5 x2 2 3x 1 2
35. y 5 3x2 2 6x 1 3
36. y 5 6x2 2 2x 1 4
37. y 5 215x2 1 10x 2 25
38. y 5 23x2 2 4x 1 8
39. y 5 9x2 2 24x 1 16
40.
★
OPEN – ENDED
41.
★
EXTENDED RESPONSE Use the rectangular prism shown.
Write a function of the form y 5 ax2 1 bx 1 c whose
graph has one x-intercept.
a. The surface area of the prism is 314 square meters. Write
an equation that you can solve to find the value of w.
8m
b. Use the discriminant to determine the number of values of
w in the equation from part (a).
c. Solve the equation. Do the value(s) of w make sense in the
context of the problem? Explain.
(w 1 4) m
CHALLENGE Find all values of k for which the equation has (a) two solutions,
(b) one solution, and (c) no solution.
42. 2x2 1 x 1 3k 5 0
43. x2 2 4kx 1 36 5 0
44. kx2 1 5x 2 16 5 0
PROBLEM SOLVING
EXAMPLE 4
45. BIOLOGY The amount y (in milliliters per gram of body mass per hour)
of oxygen consumed by a parakeet during flight can be modeled by the
function y 5 0.06x2 2 4x 1 87 where x is the speed (in kilometers per
hour) of the parakeet.
on p. 680
for Exs. 45–46
a. Use the discriminant to show that it is possible for a parakeet to
consume 25 milliliters of oxygen per gram of body mass per hour.
b. Find the speed(s) at which the parakeet consumes 25 milliliters of
oxygen per gram of body mass per hour. Round your solution(s) to
the nearest tenth.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
46. FOOD For the period 195021999, the average amount y (in pounds per
person per year) of butter consumed in the United States can be modeled
by y 5 0.0051x2 2 0.37x 1 11 where x is the number of years since 1950.
According to the model, did the butter consumption in the United States
ever reach 5 pounds per person per year? If so, in what year(s)?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
47.
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★
SHORT RESPONSE The frame of the tent shown is
defined by a rectangular base and two parabolic arches
that connect the opposite corners of the base. The graph
of y 5 20.18x2 1 1.6x models the height y (in feet) of one
of the arches x feet along the diagonal of the base. Can
a child that is 4 feet tall walk under one of the arches
without having to bend over? Explain.
5 WORKED-OUT SOLUTIONS
on p. WS1
★ 5 STANDARDIZED
TEST PRACTICE
wm
48. SCIENCE Between the months of April and September, the number y of
hours of daylight per day in Seattle, Washington, can be modeled by
y 5 20.00046x2 1 0.076x 1 13 where x is the number of days since April 1.
a. Do any of the days between April and September in Seattle have
17 hours of daylight? If so, how many?
b. Do any of the days between April and September in Seattle have
14 hours of daylight? If so, how many?
49. MULTI-STEP PROBLEM During a trampoline competition, a trampolinist
leaves the mat when her center of gravity is 6 feet above the ground. She
has an initial vertical velocity of 32 feet per second.
a. Use the vertical motion model to write an equation
that models the height h (in feet) of the center of
gravity of the trampolinist as a function of the
time t (in seconds) into her jump.
b. Does her center of gravity reach a height of 24 feet
during the jump? If so, at what time(s)?
c. On another jump, the trampolinist leaves the mat
h ft
when her center of gravity is 6 feet above the ground
and with an initial vertical velocity of 35 feet per
second. Does her center of gravity reach a height of
24 feet on this jump? If so, at what time(s)?
50. CHALLENGE Last year, a manufacturer sold backpacks for $24 each.
At this price, the manufacturer sold about 1000 backpacks per week.
A marketing analyst predicts that for every $1 reduction in the price of
the backpack, the manufacturer will sell 100 more backpacks per week.
a. Write a function that models the weekly revenue R (in dollars) that
the manufacturer will receive for x reductions of $1 in the price of
the backpack.
b. Is it possible for the manufacturer to receive a weekly revenue
of $28,000? $30,000? What is the maximum weekly revenue that
the manufacturer can receive? Explain your answers using the
discriminants of quadratic equations.
MIXED REVIEW
PREVIEW
Graph the function.
Prepare for
Lesson 10.8 in
Exs. 51–56.
51. y 5 5x 2 10 (p. 225)
1
52. y 5 }
x (p. 244)
53. y 5 }x 2 5 (p. 244)
54. y 5 5x (p. 520)
55. y 5 (0.2) x (p. 531)
56. y 5 6x2 2 3 (p. 628)
57. a 1 5 5 2 (p. 134)
58. f 2 6 5 13 (p. 134)
59. 4z 2 3 5 27 (p. 141)
60. 9w 1 4 5 241 (p. 141)
61. 2b 2 b 2 6 5 8 (p. 148)
62. 5 1 2(x 2 4) 5 9 (p. 148)
3
4
4
Solve the equation.
Solve the equation by factoring. (p. 593)
63. 2x2 2 3x 2 5 5 0
64. 4n2 1 2n 2 6 5 0
EXTRA PRACTICE for Lesson 10.7, p. 947
65. 5a2 1 21a 1 4 5 0
ONLINE QUIZ at classzone.com
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