Five-Minute Check (over Lesson 9-1)

Main Ideas and Vocabulary

Targeted TEKS

Example 1: Two Roots

Example 2: A Double Root

Example 3: No Real Roots

Example 4: Rational Roots

Example 5: Real-World Example

• quadratic equation

• roots

• zeros

• double root

• Solve quadratic equations by graphing.

• Estimate solutions of quadratic equations by graphing.

Two Roots

Solve x2 – 3x – 10 = 0 by graphing.

Graph the related function f(x) = x2 – 3x – 10.

Two Roots

Make a table of values to find other points to sketch the graph.

To solve x2 – 3x – 10 = 0 you need to know where the value of f(x) is 0. This occurs at the x-intercepts. The x-intercepts of the parabola appear to be –2 and 5.

Two Roots

Check Solve by factoring.

x2 – 3x – 10 = 0 Original equation

(x – 5)(x + 2) = 0 Factor.

x – 5 = 0 or x + 2 = 0 Zero Product Property

x = 5 x = –2 Solve for x.

Answer: The solutions of the equation are –2 and 5.

Animation: Solving Quadratic Equations By Graphing

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. {–2, 4}

B. {2, –4}

C. {2, 4}

D. {–2, –4}

Solve x2 – 2x – 8 = 0 by graphing.

A Double Root

Solve x2 – 6x = –9 by graphing.

First, rewrite the equation so one side is equal to zero.

x2 – 6x = –9 Original equation

x2 – 6x + 9 = –9 + 9 Add 9 to each side.

x2 – 6x + 9 = 0 Simplify.

A Double Root

Graph the related function f(x) = x2 – 6x + 9.

Notice that the vertex of the parabola is the x-intercept. Thus, one solution is 3. What is the other solution?

Try solving the equation by factoring.

A Double Root

x2 – 6x + 9 = 0 Original equation

Answer: The solution is 3.

(x – 3)(x – 3) = 0 Factor.

x – 3 = 0 or x – 3 = 0 Zero Product Property

x = 3 x = 3

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. {1}

B. {–1}

C. {–1, 1}

D. Ø

Solve x2 + 2x = –1 by graphing.

No Real Roots

Solve x2 + 2x + 3 = 0 by graphing.

Graph the related function f(x) = x2 + 2x + 3.

Answer: The graph has no x-intercept. Thus, there are no real number solutions for the equation.

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. {1, 5}

B. {–1, 5}

C. {5}

D. Ø

Solve x2 + 4x + 5 = 0 by graphing.

Solve x2 – 4x + 2 = 0 by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.Graph the related function f(x) = x2 – 4x + 2.

Rational Roots

Notice that the value of the function changesfrom negative to positivebetween the x values of0 and 1 and between 3and 4.

Answer: One root is between 0 and 1, and the other root is between 3 and 4.

Rational Roots

The x-intercepts of the graph are between 0 and 1 and between 3 and 4.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. one root is between 0 and 1, and the other root is between 4 and 5.

B. one root is between –1 and 0, and the other root is between 3 and 4.

C. one root is between –1 and –2, and the other root is between 3 and 4.

D. Ø

Solve x2 – 2x – 5.

MODEL ROCKETS Shelly built a model rocket for her science project. The equation y = –16t2 + 250t models the flight of the rocket, launched from ground level at a velocity of 250 feet per second, where y is the height of the rocket in feet after t seconds. For how many seconds was Shelly’s rocket in the air?

You need to find the solution of the equation 0 = –16t2

+ 250t. Use a graphing calculator to graph the related function y = –16t2 + 250t. The x-intercept is between 15 and 16 seconds.

Answer: between 15 and 16 seconds.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. between 7 and 8 seconds

B. between 6 and 7 seconds

C. between 8 and 9 seconds

D. between 0 and 1 second

GOLF Martin hits a golf ball with an upward velocity of 120 feet per second. The function y = –16t2 + 120t models the flight of the golf ball, hit at ground level, where y is the height of the ball in feet after t seconds. How long was the golf ball in the air?