CA ADV Algebra Standard 08
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. A toy rocket is launched from the ground level with an initial vertical velocity of 96 ft/s. After how many seconds will the rocket hit the ground?
a.

6 seconds

b.

0 seconds or 6 seconds

c.

0 seconds

d.

seconds

____ 2. Find the roots of the equation by factoring.
____ 3. Solve the equation .
____ 4. Complete the square for the expression _____. Write the resulting expression as a binomial squared.
____ 5. Solve the equation by completing the square.
a.

x = –1 or x = 3

b.

x = 2 or x = –6

c.

x = 1 or x = –3

d.

x = 2 or x = –2

____ 6. Solve the equation .
____ 7. Find the zeros of by using the Quadratic Formula.
____ 8. Find the zeros of by using the Quadratic Formula.
____ 9. Find the number and type of solutions for .
a.

The equation has two real solutions.

b.

Cannot determine without graphing.

c.

The equation has one real solution.

d.

The equation has two nonreal complex solutions.

____ 10. During the eruption of Mount St. Helens in 1980, debris was ejected at a speed of over 440 feet per second (300 miles per hour). The height in feet of a rock ejected at angle of 75º is given by the equation , where t is the time in seconds after the eruption. The rock’s horizontal distance in feet from the point of ejection is given by . Assuming the elevation of the surrounding countryside is 0 feet, what is the horizontal distance from the point of ejection to the where the rock would have landed? Round your answer to the nearest foot.
a.

1,117 ft

b.

2,234 ft

c.

4,467 ft.

d.

8,932 ft

____ 11. The daily profit P for a cake bakery can be modeled by the function . What should the price of a cake be to provide a daily profit of at least $600? Round your answer(s) to the nearest dollar.
____ 12. Solve .
Numeric Response
13. Find the positive root of .
14. Find the positive root of the equation by completing the square.
CA ADV Algebra Standard 08
Answer Section
MULTIPLE CHOICE
1. ANS: A

Write the general projectile function.


Substitute 96 for and 0 for .

The toy rocket will hit the ground when its height is zero.

Set equal to 0.


Factor. The GCF is –16t.

or

Apply the Zero Product Property.

or

Solve each equation.

The toy rocket will hit the ground in 6 seconds. Notice that the height is also zero when , the instant that the toy rocket is launched.

Feedback

A

Correct!

B

The height is also zero when t = 0, but this is the instant that the toy rocket is launched.

C

There are two solutions to the equation, and zero is one of them. But t = 0 is the instant that the toy rocket is launched.

D

Your substitution for the initial velocity and height are reversed. Also, the leading coefficient is negative.

PTS: 1 DIF: Average REF: Page 335 OBJ: 53.3 Application
NAT: 12.5.4.a STA: 2A8.0
TOP: 53 Solving Quadratic Equations by Graphing and Factoring
2. ANS: B
Rewrite the equation in standard form, factor out the GCF, and then factor the perfect square trinomial.

Feedback

A

Get all of the terms on the same side of the equation, then factor.

B

Correct!

C

Get all of the terms on the same side of the equation, then factor.

D

Be careful with your plus and minus signs when factoring.

PTS: 1 DIF: Average REF: Page 336
OBJ: 53.4 Finding Roots by Using Special Factors NAT: 12.5.4.a
STA: 2A8.0 TOP: 53 Solving Quadratic Equations by Graphing and Factoring
3. ANS: C

Factor the perfect square trinomial.


Take the square root of both sides.


Add 5 to each side.


Simplify.


Feedback

A

Does the right side of the equation have only a negative square root?

B

You switched the number to the left of the radical sign with the number under the radical sign.

C

Correct!

D

Does the right side of the equation have only a positive square root?

PTS: 1 DIF: Average REF: Page 341
OBJ: 54.1 Solving Equations by Using the Square Root Property
NAT: 12.5.4.a STA: 2A8.0 TOP: 54 Completing the Square
4. ANS: A

Feedback

A

Correct!

B

Add (b/2)^2 to the given expression, then factor.

C

Is b positive or negative?

D

Add (b/2)^2 to the given expression, then factor.

PTS: 1 DIF: Average REF: Page 342 OBJ: 54.2 Completing the Square
NAT: 12.5.3.d STA: 2A8.0 TOP: 54 Completing the Square
5. ANS: C

Feedback

A

Remember to subtract b/2 from both sides.

B

Remember to divide by 2.

C

Correct!

D

Remember to solve for x after taking the square root.

PTS: 1 DIF: Basic REF: Page 343
OBJ: 54.3 Solving a Quadratic Equation by Completing the Square
NAT: 12.5.4.a STA: 2A8.0 TOP: 54 Completing the Square
6. ANS: A

Add to both sides.


Divide both sides by 2.


Take square roots.


Express in terms of i.


Feedback

A

Correct!

B

The square root of a negative number is an imaginary number.

C

The square root of a negative number is not a sum or a difference. It is a product of i and the square root of the opposite number.

D

The square root of a negative number is not a sum. It is a product of i and the square root of the opposite number.

PTS: 1 DIF: Average REF: Page 351
OBJ: 55.2 Solving a Quadratic Equation with Imaginary Solutions
STA: 2A8.0 TOP: 55 Complex Numbers and Roots
7. ANS: A

Set .


Write the Quadratic Formula.


Substitute 1 for a, 7 for b, and 9 for c.


Simplify.


Write in simplest form.


Feedback

A

Correct!

B

Rewrite the equation in standard form to get the values of a, b, and c for the Quadratic Formula.

C

Set f(x) = 0, and then use the Quadratic Formula.

D

Use the Quadratic Formula.

PTS: 1 DIF: Average REF: Page 357
OBJ: 56.1 Quadratic Functions with Real Zeros NAT: 12.5.4.a
STA: 2A8.0 TOP: 56 The Quadratic Formula
8. ANS: D

Set .


Write the Quadratic Formula.


Substitute 4 for a, –1 for b and 5 for c.


Simplify.


Write in terms of i.


Feedback

A

b = –1, and –1 squared is 1.

B

The square root of a negative number is equal to i times the square root of the opposite number.

C

The denominator of the Quadratic Formula is 2 times a.

D

Correct!

PTS: 1 DIF: Average REF: Page 357
OBJ: 56.2 Quadratic Functions with Complex Zeros STA: 2A8.0
TOP: 56 The Quadratic Formula
9. ANS: A

Make sure the equation is in standard form, ax^{2} + bx + c = 0.






Evaluate the discriminant.

=


=


The discriminant is positive. The equation has two real solutions.

Feedback

A

Correct!

B

Put the equation in standard form. Is the discriminant positive, negative, or zero?

C

Put the equation in standard form. Is the discriminant positive, negative, or zero?

D

Put the equation in standard form. Is the discriminant positive, negative, or zero?

PTS: 1 DIF: Average REF: Page 358
OBJ: 56.3 Analyzing Quadratic Equations by Using the Discriminant
NAT: 12.5.4.a STA: 2A8.0 TOP: 56 The Quadratic Formula
10. ANS: C
Use the first equation to determine how long it took the rock to hit the ground.



Set y equal to 0.


Use the Quadratic Formula.


Substitute for a, b, and c.





The time cannot be negative, so the rocks hit the ground approximately 39.5 seconds after it was ejected. Find the horizontal distance the rock traveled during this time.



Substitute 39.5 for t.


Simplify.


Feedback

A

Set y = 0 and solve for t using the Quadratic Formula. Then find the horizontal distance x(t).

B

Set y = 0 and solve for t using the Quadratic Formula. Then find the horizontal distance x(t).

C

Correct!

D

Set y = 0 and solve for t using the Quadratic Formula. Then find the horizontal distance x(t).

PTS: 1 DIF: Average REF: Page 359 OBJ: 56.4 Application
NAT: 12.5.4.a STA: 2A8.0 TOP: 56 The Quadratic Formula
11. ANS: D
The profit must be at least $600.
Find the critical values by solving the related equation.
or
Use a number line and test an xvalue in each of the three regions created by the critical points.
Try x = 10:
–15(10)^{2} + 330(10) – 815
= 985
985 ³ 600

Try x = 18:
–15(20)^{2} + 330(20) – 815
= –215
–215 is not ³ 600

Try x = 4:
–15(4)^{2} + 330(4) – 815 = 265
265 is not ³ 600

Round.

Feedback

A

If the price is 0 dollars then the profit is –$815. Use a number line and critical values to find the answer.

B

If the price is 0 dollars then the profit is –$815. Use a number line and critical values to find the answer.

C

If the price is $19 then the profit is only $40. Use a number line and critical values to find the answer.

D

Correct!

PTS: 1 DIF: Average REF: Page 368 OBJ: 57.4 ProblemSolving Application
NAT: 12.5.4.a STA: 2A8.0 TOP: 57 Solving Quadratic Inequalities
12. ANS: A



Factor.

or

Use the Zero Product Property.

or

Solve each equation.


Feedback

A

Correct!

B

Check the signs.

C

Factor, and then solve each equation.

D

Factor, and then solve each equation.

PTS: 1 DIF: Average OBJ: Solving Using Different Methods
NAT: 12.5.4.a STA: 2A8.0 TOP: The Quadratic Formula
NUMERIC RESPONSE
13. ANS: 5
PTS: 1 DIF: Advanced NAT: 12.5.4.a STA: 2A8.0
TOP: 53 Solving Quadratic Equations by Graphing and Factoring
14. ANS: 1.25
PTS: 1 DIF: Advanced NAT: 12.5.4.a STA: 2A8.0
TOP: 54 Completing the Square 