5. Example 1 Use the discriminant
Equation Discriminant Number of
solutions
a. 2x2 + 6x + 5 = 0 62 – 4(2 ) (5 ) = –4 No solution
b. x2 – 7 = 0 02 – 4(1 ) ( – 7 ) = 28 Two solutions
c. 4x2 – 12x + 9 = 0 ( –12 )2 – 4(4 ) (9) = 0 One solution
6. Example 2 Multiple Choice Practice
Which statement best explains why there is only one
real solution to the quadratic equation 9x2 + 6x + 1 = 0?
The value of ( 6 )2 – 4 • 9 • 1 is positive.
The value of ( 6 )2 – 4 • 9 • 1 is equal to 0.
The value of ( 6 )2 – 4 • 9 • 1 is negative.
The value of ( 6 )2 – 4 • 9 • 1 is not a perfect
square.
7. Example 2 Multiple Choice Practice
SOLUTION
Find the value of the discriminant.
b2 – 4 • a • c = ( 6 )2 – 4 • 9 • 1 = 36 – 36 = 0
The discriminant is zero, so the equation has one real
solution.
ANSWER The correct answer is B.
8. Example 3 Find the number of x-intercepts
Find the number of x-intercepts of the graph of
y = x2 – 3x – 10.
SOLUTION
Find the number of solutions of the equation
0 = x2 – 3x – 10.
b2 – 4ac = ( – 3)2 – 4(1 ) ( –10 ) Substitute 1 for a, – 3 for b,
and –10 for c.
= 49 Simplify.
The discriminant is positive, so the equation has two
solutions. This means that the graph of y = x2 – 3x – 10
has two x-intercepts.
9. Example 3 Find the number of x-intercepts
CHECK You can use a graphing calculator to check
the answer. Notice that the graph of
y = x2 – 3x – 10 has two x-intercepts.
You can also use factoring to check the answer.
Because x2 – 3x – 10 = ( x – 5 ) ( x + 2 ), the
graph of y = x2 – 3x – 10 crosses the x-axis at
x – 5 = 0, or x = 5, and at x + 2 = 0, or x = – 2.
10. 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 = – 0.006x2 + 1.2x + 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?
11. Example 4 Solve a multi-step problem
SOLUTION
STEP 1 Write a quadratic equation. You want to know
whether the water arc reaches a height of 50
feet, so let y = 50. Then write the quadratic
equation in standard form.
y = – 0.006x2 + 1.2x + 10 Write given equation.
50 = – 0.006x2 + 1.2x + 10 Substitute 50 for y.
0 = – 0.006x2 + 1.2x – 40 Subtract 50 from each side.
STEP 2 Find the value of the discriminant of
0 = – 0.006x2 + 1.2x – 40.
12. Example 4 Solve a multi-step problem
b2 – 4ac = ( 1.2)2 – 4 ( – 0.006 ) ( – 40 ) a = – 0.006, b = 1.2,
c = – 40
= 0.48 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 = – 0.006x2 + 1.2x – 40 to
find the distance from the north shore where
the water arc is 50 feet above the water.
13. Example 4 Solve a multi-step problem
–b +
– b2 – 4ac
x = Quadratic formula
2a
– 1.2 +
– 0.48
= Substitute values in
2 ( – 0.006 ) the quadratic formula.
x ≈ 42 or 158 Use a calculator.
ANSWER
The water arc is 50 feet above the water about 42 feet
from the north shore and about 158 feet from the north
shore.
14. 10.8 Warm-Up
Tell whether the equation has 2 solutions, one solution,
or no solution.
1. x 2 + 4x + 3 = 0
2. 2x 2 - 5x + 6 = 0
3. -x 2 + 2x =1
Find the number of x-intercepts of the graph of the
function.
4. y = x 2 +10x + 25
5. y = x 2 - 9x
