10.7

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Interpret the Discriminant

10.7 Warm-Up

Evaluate the expression.

1. 8x2 when x = 4

ANSWER 128

ANSWER –1

2. –4 + x when x = 9√

ANSWER 18.5 ft

A ball is kicked into the air from a height of 4.5 feet with an initial velocity of 30 feet per second. What is the height of the ball after 1 second?

3.

10.7 Example 1

Number of solutions

Equationax2 + bx + c = 0

Discriminantb2 – 4ac

a. 2x2 + 6x + 5 = 0

b. x2 – 7 = 0

c. 4x2 – 12x + 9 = 0

62 – 4(2)(5) = –4 No solution

02 – 4(1)(– 7) = 28 Two solutions

(–12)2 –4(4)(9) = 0 One solution

Use the discriminant to tell whether the equation has two solutions, one solution, or no solution.

10.7 Example 2

Tell whether the equation 3x2 – 7 = 2x has two solutions, one solution, or no solution.

SOLUTION

STEP 1 Write the equation in standard form.

3x2 – 7 = 2x

3x2 – 2x – 7 = 0

Write equation.

Subtract 2x from each side.

10.7 Example 2

STEP 2 Find the value of the discriminant.

b2 – 4ac = (–2)2 – 4(3)(–7)

= 88

Substitute 3 for a, – 2 for b,

and –7 for c.

Simplify.

The discriminant is positive, so the equation has two solutions.

ANSWER

10.7 Guided Practice

Tell whether the equation has two solutions, one

solution, or no solution.1. x2 + 4x + 3 = 0 ANSWER two solutions

2. 2x2 – 5x + 6 = 0 ANSWER no solution

3. –x2 + 2x = 1 ANSWER one solution

10.7 Example 3

Find the number of x-intercepts of the graph of y = x2 + 5x + 8.

b2 – 4ac = (5)2 – 4(1)(8) Substitute 1 for a, 5 for b, and 8 for c.

SOLUTION

Find the number of solutions of the equation 0 = x2 + 5x + 8.

Simplify.–7=

The discriminant is negative, so the equation has no solution. This means that the graph of y = x2 + 5x + 8 has no x-intercepts.

ANSWER

10.7 Example 3

CHECK

You can use a graphing calculator to check the answer. Notice that the graph of y = x2 + 5x + 8 has no x-intercepts.

10.7 Guided Practice

Find the number of x – intercepts of the graph of the function.

4. y = x2 + 10x + 25 ANSWER 1

5. y = x2 – 9x ANSWER 2

6. y = x2 + 2x 4 ANSWER 0

10.7 Example 4

FOUNTAINSThe 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?

10.7 Example 4

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.

Write given equation.y 0.006x2 1.2x 10 = – + +

50 0.006x2 1.2x 10= – + + Substitute 50 for y.

Subtract 50 from each side.0 0.006x2 1.2x 40= – + –

10.7 Example 4

b2 4ac = (1.2)2 4(–0.006)(–40)– – = 0.48

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 2

Find the value of the discriminant of0 = – 0.006x2 + 1.2x – 40.

a = –0.006, b = 1.2, c = –40

Simplify.

10.7 Example 4

–x = b 2 4ac– ±b

2aQuadratic formula

= – ±

1.2 0.48

2(–0.006)

Substitute values in the quadratic formula.

Use a calculator.x 42 or x 158

STEP 4Solve the equation to find the distance from the north shore to where the water arc is 50 feet above the water.

0 = – 0.006x2 + 1.2x – 40

The water arc is 50 feet above the water about 42 feet from the north shore and about 158 feet from the north shore.

ANSWER

10.7 Guided Practice

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?

Yes; 100 ft

ANSWER

10.7 Lesson Quiz

1. 4b2 + 2b – 5 = 0

Tell whether the equation has two solutions, one solution, or no solutions.

2. 2g2 + 8g = – 11 ANSWER no solutions

ANSWER two solutions

3. y = x2 + 14x + 49 ANSWER one

y = x2 + 14x + 504. ANSWER none

Find the number of x-intercepts of the graphs of the equation.

10.7 Lesson Quiz

5. The graph of y = – 0.2x2 + 3.5x models the height of one of the arches at the entrance to a parking structure. Can a truck that is 20 feet high fit under the arch?

The value of the discriminate of 0 = – 0.2x2 + 3.5x – 20 is negative, so there is no solution.The truck will not fit under the arch.

ANSWER