Page 1 of 6
9.7
Using the Discriminant
Can you throw a stick high enough?
Goal
Use the discriminant to
determine the number of
solutions of a quadratic
equation.
One way that campers protect food
from bears is to hang it from a high
tree branch. In the example on
page 545, you will determine
whether a stick and a rope were
thrown fast enough to go over a
tree branch.
Key Words
• discriminant
In the quadratic formula, the expression inside the radical is the discriminant.
b b2
4ac
x 2a
Discriminant
The discriminant b2 4ac of a quadratic equation can be used to find the number
of solutions of the quadratic equation.
Student Help
THE NUMBER OF SOLUTIONS OF A QUADRATIC EQUATION
STUDY TIP
Recall that positive
real numbers have two
square roots, zero has
only one square root,
negative numbers have
no real square roots.
Consider the quadratic equation ax2 bx c 0.
• If the value of b 2 4ac is positive, then the equation has two
solutions.
• If the value of b 2 4ac is zero, then the equation has one
solution.
• If the value of b 2 4ac is negative, then the equation has no real
solution.
EXAMPLE
1
Find the Number of Solutions
Find the value of the discriminant. Then use the value to determine whether
x2 3x 4 0 has two solutions, one solution, or no real solution.
Solution Use the standard form of a quadratic equation, ax2 bx c 0,
to identify the coefficients.
x2 3x 4 0
b2 4ac (3)2 4(1)(4)
ANSWER 540
Chapter 9
Identify a 1, b 3, c 4.
Substitute values for a, b, and c.
9 16
Simplify.
25
Discriminant is positive.
The discriminant is positive, so the equation has two solutions.
Quadratic Equations and Functions
Page 2 of 6
EXAMPLE
2
Find the Number of Solutions
Find the value of the discriminant. Then use the value to determine whether the
equation has two solutions, one solution, or no real solution.
a. x2 2x 1 0
b. 2x2 2x 3 0
Solution
a. x2 2x 1 0
Identify a 1, b 2, c 1.
b 4ac (2) 4(1)(1)
2
ANSWER 2
Substitute values for a, b, and c.
44
Simplify.
0
Discriminant is zero.
The discriminant is zero, so the equation has one solution.
b. 2x 2x 3 0
2
Identify a 2, b 2, c 3.
b2 4ac (2)2 4(2)(3)
ANSWER Substitute values for a, b, and c.
4 24
Simplify.
20
Discriminant is negative.
The discriminant is negative, so the equation has no
real solution.
Find the Number of Solutions
Find the value of the discriminant. Then use the value to determine
whether the equation has two solutions, one solution, or no real solution.
1. x2 3x 4 0
2. x2 4x 4 0
3. x2 5x 4 0
Because each solution of ax2 bx c 0 represents an x-intercept of
y ax2 bx c, you can use the discriminant to determine the number of
times the graph of a quadratic function intersects the x-axis.
EXAMPLE
3
Find the Number of x-Intercepts
Determine whether the graph of y x2 2x 2 will intersect the x-axis in
zero, one, or two points.
Solution
Let y 0. Then find the value of the discriminant of x2 2x 2 0.
x2 2x 2 0
Identify a 1, b 2, c 2.
b 4ac (2) 4(1)(2)
2
ANSWER 2
Substitute values for a, b, and c.
48
Simplify.
12
Discriminant is positive.
The discriminant is positive, so the equation has two solutions and
the graph will intersect the x-axis in two points.
9.7
Using the Discriminant
541
Page 3 of 6
EXAMPLE
Student Help
MORE EXAMPLES
NE
ER T
INT
More examples
are available at
www.mcdougallittell.com
Find the Number of x-Intercepts
4
Determine whether the graph of the function will intersect the x-axis in zero,
one, or two points.
a. y x2 2x 1
b. y x2 2x 3
Solution
a. Let y 0. Then find the value of the discriminant of x2 2x 1 0.
x2 2x 1 0
Identify a 1, b 2, c 1.
b2 4ac (2)2 4(1)(1)
ANSWER Substitute values for a, b, and c.
44
Simplify.
0
Discriminant is zero.
The discriminant is zero, so the equation has one solution and
the graph will intersect the x-axis in one point.
b. Let y 0. Then find the value of the discriminant of x2 2x 3 0.
x2 2x 3 0
Identify a 1, b 2, c 3.
b 4ac (2) 4(1)(3)
2
ANSWER EXAMPLE
Student Help
LOOK BACK
For help with sketching
the graph of a quadratic
function, see p. 521.
2
Substitute values for a, b, and c.
4 12
Simplify.
8
Discriminant is negative.
The discriminant is negative, so the equation has no real solution
and the graph will intersect the x-axis in zero points.
5
Change the Value of c
Sketch the graphs of the functions in Examples 3 and 4 to check the number of
x-intercepts of y x2 2x c. What effect does changing the value of c have
on the graph?
Solution By changing the value of c, you
can move the graph of y x2 2x c up
or down in the coordinate plane.
y
4
c
a. y x2 2x 2
b. y x2 2x 1
b
4
c. y x2 2x 3
If the graph is moved high enough, it will not have
an x-intercept and the equation x2 2x c 0
will have no real solution.
1
1
a
Find the Number of x-Intercepts
Find the number of x-intercepts of the graph of the function.
4. y x2 4x 3
542
Chapter 9
Quadratic Equations and Functions
5. y x2 4x 4
6. y x2 4x 5
2
x
Page 4 of 6
9.7 Exercises
Guided Practice
Vocabulary Check
1. Write the quadratic formula and circle the part that is the discriminant.
2. What can the discriminant tell you about a quadratic equation?
3. Describe how the graphs of y 4x2, y 4x2 3, and y 4x2 6 are alike
and how they are different.
Skill Check
Use the discriminant to determine whether the quadratic equation has
two solutions, one solution, or no real solution.
4. 3x2 3x 5 0
5. 3x2 6x 3 0
6. x2 5x 10 0
Give the letter of the graph that matches the value of the discriminant.
7. b2 4ac 2
8. b2 4ac 0
9. b2 4ac 3
A.
B.
C.
y
y
x
y
x
x
Determine whether the graph of the function will intersect the x-axis in
zero, one, or two points.
10. y x2 2x 4
11. y x2 3x 5
12. y 6x 3 3x2
Practice and Applications
WRITING THE DISCRIMINANT Find the discriminant of the quadratic
equation.
Student Help
HOMEWORK HELP
Example 1: Exs. 13–33
Example 2: Exs. 13–33
Example 3: Exs. 38–43
Example 4: Exs. 38–43
Example 5: Exs. 44–46
13. 2x2 5x 3 0
14. 3x2 6x 8 0
15. x2 10 0
16. 5x2 3x 12
17. 2x2 8x 8
18. 7 5x2 9x x
19. x 7x2 4
20. 2x x2 x
21. 2 x2 4x2
USING THE DISCRIMINANT Determine whether the equation has two
solutions, one solution, or no real solution.
22. x2 3x 2 0
23. 2x2 4x 3 0
24. 3x2 5x 1 0
25. 2x2 3x 2 0
26. x2 2x 4 0
27. 6x2 2x 4 0
28. 3x2 6x 3 0
29. 4x2 5x 1 0
30. 5x2 6x 6 0
1
31. x2 x 3 0
2
1
32. x2 2x 4 0
4
4
33. 5x2 4x 0
5
9.7
Using the Discriminant
543
Page 5 of 6
34. ERROR ANALYSIS For the
equation 3x2 4x 2 0,
b2 – 4ac
=(4)2 – 4(3)(–2)
= 16 – 24
= –8
find and correct the error.
INTERPRETING THE DISCRIMINANT In Exercises 35–37, consider the
quadratic equation y 2x2 6x 3.
35. Evaluate the discriminant.
36. How many solutions does the equation have?
37. What does the discriminant tell you about the graph of y 2x2 6x 3?
NUMBER OF X-INTERCEPTS Determine whether the graph of the
function will intersect the x-axis in zero, one, or two points.
38. y 2x2 3x 2
39. y x2 2x 4
40. y 2x2 4x 2
41. y 2x2 2x 6
42. y 5x2 2x 3
43. y 3x2 6x 3
CHANGING THE VALUE OF C Match the function with its graph.
44. y x2 2x 1
A.
45. y x2 2x 3
B.
y
2
y
2
4 x
4
6
Careers
C.
y
2
6
46. y x2 2x 3
2
2
2
6
x
6
2
2
x
6
FINANCIAL ANALYSIS In Exercises 47–49, use a graphing calculator
and the following information.
FINANCIAL ANALYSTS
INT
use mathematical models to
help analyze and predict a
company’s future earnings.
NE
ER T
More about financial
analysts is available
at www.mcdougallittell.com
544
Chapter 9
Software Company Profits
Profit
(millions
of dollars)
A software company’s net profit
for each year from 1993 to 1998
lead a financial analyst to model
the company’s net profit by
P 6.84t2 3.76t 9.29,
where P is the profit in millions
of dollars and t is the number of
years since 1993. In 1993 the
net profit was approximately
9.29 million dollars (t 0).
P
400
200
0
0
2
4
6
8
Years since 1993
t
47. Give the domain and range of the function for 1993 through 1998.
48. Use the graph to predict whether the net profit will reach 650 million dollars.
49. Use a graphing calculator to estimate how many years it will take for the
company’s net profit to reach 475 million dollars according to the model.
Quadratic Equations and Functions
Page 6 of 6
Using the Discriminant
EXAMPLE
Camping
You and a friend want to get a rope over a tree branch that is
20 feet high. Your friend attaches a stick to the rope and throws the stick
upward with an initial velocity of 29 feet per second. You then throw it with
an initial velocity of 32 feet per second. Both throws have an initial height of
6 feet. Will the stick reach the branch each time it is thrown?
CAMPING
Solution
Use a vertical motion model for an object that is thrown: h 16t2 vt s,
where h is the height you want to reach, t is the time in motion, v is the initial
velocity, and s is the initial height.
h 16t2 vt s
h 16t2 vt s
20 16t2 29t 6
20 16t2 32t 6
0 16t2 29t 14
CAMPING Bears have an
excellent sense of smell that
often leads them to campsites
in search of food. Campers
can hang a food sack from a
high tree branch to keep it
away from bears.
0 16t2 32t 14
b2 4ac (29)2 4(16)(14)
b2 4ac (32)2 4(16)(14)
The discriminant is 55.
The discriminant is 128.
ANSWER ANSWER The discriminant is
negative. The stick
thrown by your
friend will not reach
the branch.
The discriminant is
positive. The stick
thrown by you will
reach the branch.
50. BASKETBALL You can jump with an initial velocity of 12 feet per second.
You need to jump 2.2 feet to dunk a basketball. Use the vertical motion
model h 16t2 vt s to find if you can dunk the ball. Justify your
answer.
Standardized Test
Practice
51. MULTIPLE CHOICE For which value of c will 3x2 6x c 0 not have
a real solution?
A
c < 3
B
c 3
C
D
c > 3
c3
52. MULTIPLE CHOICE How many real solutions does x2 10x 25 0 have?
F
Mixed Review
No solutions
G
One solution
H
Two solutions
J
Many solutions
SOLVING AND GRAPHING INEQUALITIES Solve the inequality. Then
graph the solution. (Lesson 6.4)
53. 2 ≤ x 1 < 5
54. 8 > 2x > 4
55. 12 < 2x 6 < 4
GRAPHING LINEAR INEQUALITIES Graph the inequality. (Lesson 6.8)
56. 3x y ≤ 9
Maintaining Skills
58. 2x y ≥ 4
57. y 4x < 0
MULTIPLYING DECIMALS Find the product. (Skills Review p. 759)
59. 3 0.02
60. 0.7 0.8
61. 0.1 0.1
62. 0.05 0.003
63. 0.09 0.02
64. 0.06 0.0004
9.7
Using the Discriminant
545