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Math Problems
Algebra 2
Conjugate root theorems
g
(
x
)
=
{
−
8
,
−
9
≤
x
<
3
4
,
3
≤
x
≤
9
g(x)=\left\{\begin{array}{ll}-8 & , \quad-9 \leq x<3 \\4 & , \quad 3 \leq x \leq 9\end{array}\right.
g
(
x
)
=
{
−
8
4
,
−
9
≤
x
<
3
,
3
≤
x
≤
9
\newline
What is the graph of
g
g
g
?
\newline
Choose
1
1
1
answer:
\newline
(A)
A
A
A
\newline
(B)
B
B
B
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h
(
x
)
=
{
2
−
x
,
−
7
≤
x
≤
5
3
x
−
21
,
5
<
x
≤
9
h(x)=\left\{\begin{array}{ll} 2-x & , \quad-7 \leq x \leq 5 \\ 3 x-21 & , \quad 5<x \leq 9 \end{array}\right.
h
(
x
)
=
{
2
−
x
3
x
−
21
,
−
7
≤
x
≤
5
,
5
<
x
≤
9
\newline
What is the graph of
h
h
h
?
\newline
Choose
1
1
1
answer:
\newline
(A)
A
A
A
\newline
A
y
A \quad y
A
y
\newline
(B)
\newline
B
B
B
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The graph of a sinusoidal function intersects its midline at
(
0
,
−
6
)
(0,-6)
(
0
,
−
6
)
and then has a minimum point at
(
2.5
,
−
9
)
(2.5,-9)
(
2.5
,
−
9
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
f(x)=
f
(
x
)
=
□
\square
□
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Choose the word that makes this sentence true.
\newline
A parallelogram is ____ a quadrilateral.
\newline
Choices:
\newline
(A)always
\newline
(B)sometimes
\newline
(C)never
\newline
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Choose the word that makes this sentence true.
\newline
A rhombus is ____ a parallelogram.
\newline
Choices:
\newline
(A)always
\newline
(B)sometimes
\newline
(C)never
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Choose the word that makes this sentence true.
\newline
A rectangle is ____ a square.
\newline
Choices:
\newline
(A) always
\newline
(B) sometimes
\newline
(C) never
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Choose the word that makes this sentence true.
\newline
A kite is ____ a quadrilateral.
\newline
Choices:
\newline
(A) always
\newline
(B) sometimes
\newline
(C) never
Get tutor help
Choose the word that makes this sentence true.
\newline
A rhombus is ____ a rectangle.
\newline
Choices:
\newline
(A) always
\newline
(B) sometimes
\newline
(C) never
Get tutor help
Choose the word that makes this sentence true.
\newline
A rectangle is ____ a quadrilateral.
\newline
Choices:
\newline
(A) always
\newline
(B) sometimes
\newline
(C) never
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Choose the word that makes this sentence true.
\newline
A square is ____ a parallelogram.
\newline
Choices:
\newline
(A) always
\newline
(B) sometimes
\newline
(C) never
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Let
g
(
x
)
=
x
2
−
x
−
12
x
−
4
g(x)=\frac{x^{2}-x-12}{x-4}
g
(
x
)
=
x
−
4
x
2
−
x
−
12
when
x
≠
4
x\neq 4
x
=
4
.
\newline
g
g
g
is continuous for all real numbers.
\newline
Find
g
(
4
)
g(4)
g
(
4
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
4
-4
−
4
\newline
(B)
7
7
7
\newline
(C)
−
3
-3
−
3
\newline
(D)
4
4
4
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Solve the following system of equations using an inverse matrix. You must also indicate the inverse matrix,
A
−
1
A^{-1}
A
−
1
, that was used to solve the system. You may optionally write the inverse matrix with a scalar coefficient.
\newline
−
2
x
+
7
y
=
9
-2x+7y=9
−
2
x
+
7
y
=
9
\newline
−
3
x
+
8
y
=
4
-3x+8y=4
−
3
x
+
8
y
=
4
\newline
A
−
1
=
□
A^{-1}=\square
A
−
1
=
□
\newline
x
=
x=
x
=
□
\square
□
\newline
y
=
□
y=\square
y
=
□
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\newline
Given the function
f
(
x
)
=
(
x
+
4
)
2
(
x
−
2
)
2
(
x
−
1
)
2
(
x
−
3
)
f(x)=\frac{(x+4)^{2}(x-2)}{2(x-1)^{2}(x-3)}
f
(
x
)
=
2
(
x
−
1
)
2
(
x
−
3
)
(
x
+
4
)
2
(
x
−
2
)
, use the characteristics of polynomials and rational functions to describe its behavior and sketch the function.
\newline
Enter the exact answers.
\newline
Enter the intercepts as points,
(
a
,
b
)
(a, b)
(
a
,
b
)
. Enter the
x
x
x
-intercepts in increasing order of the
x
x
x
-coordinate.
\newline
The
x
x
x
-intercepts are
□
\square
□
and
□
\square
□
\newline
The
y
y
y
-intercept is
□
\square
□
\newline
The fields below accept a list of numbers or formulas separated by semicolons (e.g.
2
;
4
;
6
2 ; 4 ; 6
2
;
4
;
6
or
(
a
,
b
)
(a, b)
(
a
,
b
)
0
0
0
. The order of the list does not matter.
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Find all solutions of the equation below.
\newline
4
3
x
+
1
=
7
x
4^{3 \mathrm{x}+1}=7^{x}
4
3
x
+
1
=
7
x
\newline
The solution(s) is/are
x
=
x=
x
=
□
\square
□
\newline
(Simplify your answer. Use a comma to separate answers as needed. Round to four decimal places as needed.)
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Find the volume of the solid generated by revolving the region bounded by the graphs of
y
=
x
2
+
3
y=x^{2}+3
y
=
x
2
+
3
and
y
=
x
+
9
y=x+9
y
=
x
+
9
about the
x
x
x
-axis.
\newline
The volume of the solid is
□
\square
□
cubic units.
\newline
(Type an exact answer, using
π
\pi
π
as needed.)
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Determine the qualities of the given set. (Select all that apply.)
\newline
{
(
x
,
y
)
∣
9
<
x
2
+
y
2
<
25
}
\left\{(x, y) \mid 9<x^{2}+y^{2}<25\right\}
{
(
x
,
y
)
∣
9
<
x
2
+
y
2
<
25
}
\newline
□
\square
□
open
\newline
□
\square
□
connected
\newline
□
\square
□
simply-connected
\newline
□
\square
□
none of the above
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Use the error formula to find
n
n
n
such that the error is less than or equal to
0.00001
0.00001
0.00001
using the Trapezoidal Rule.
\newline
∫
0
π
2
sin
x
d
x
\int_{0}^{\frac{\pi}{2}}\sin x\,dx
∫
0
2
π
sin
x
d
x
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The Apollo
11
11
11
mission is famous for putting the first man on the moon. Jenny informed her friends that the Apollo
11
11
11
lunar module was on the surface of the moon for
8
×
1
0
7
8 \times 10^7
8
×
1
0
7
milliseconds. What would be the most appropriate unit of time for Jenny to use instead of milliseconds?
\newline
Choices:
\newline
(A) seconds
\newline
(B) minutes
\newline
(C) hours
\newline
(D) weeks
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Solve for
x
x
x
. Express your answer in simplest radical form if necessary.
\newline
x
2
=
4
x^{2}=4
x
2
=
4
\newline
Answer:
x
=
x=
x
=
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Solve for
x
x
x
. Express your answer in simplest radical form if necessary.
\newline
x
2
=
36
x^{2}=36
x
2
=
36
\newline
Answer:
x
=
x=
x
=
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Express as a single fraction in its simplest form.
\newline
1
+
y
−
y
+
2
y
−
1
1+y-\frac{y+2}{y-1}
1
+
y
−
y
−
1
y
+
2
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Use the rational zeros theorem to determine the potential rational zeros of the polynomial function. Do not find the zeros.
\newline
f
(
x
)
=
x
3
−
6
x
−
81
f(x)=x^{3}-6 x-81
f
(
x
)
=
x
3
−
6
x
−
81
\newline
List the possible potential rational zeros of the polynomial function.
\newline
□
\square
□
(Type an integer or a fraction. Use a comma to separate answers as needed.)
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Use the rational zeros theorem to determine the potential rational zeros of the polynomial function. Do not find the zeros.
\newline
f
(
x
)
=
x
3
−
6
x
−
81
f(x)=x^{3}-6 x-81
f
(
x
)
=
x
3
−
6
x
−
81
\newline
List the possible potential rational zeros of the polynomial function.
\newline
□
\square
□
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If the product of the zeroes of the quadratic polynomial
p
(
x
)
=
a
x
2
−
6
x
−
6
p(x)=a x^{2}-6 x-6
p
(
x
)
=
a
x
2
−
6
x
−
6
is
4
4
4
, then find the value of
a
a
a
. Also, find the sum of the zeroes of the polynomial.
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If the product of the zeroes of the quadratic polynomial
p
(
x
)
=
a
x
2
−
6
x
−
6
p(x)=a x^{2}-6 x-6
p
(
x
)
=
a
x
2
−
6
x
−
6
is
4
4
4
, then find the value of
a
a
a
. Also, find the sum of the zeroes of the polynomial.
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Find the value of
k
k
k
such that
3
x
2
+
2
k
x
−
k
−
5
3 x^{2}+2 k x-k-5
3
x
2
+
2
k
x
−
k
−
5
has the sum of the zeroes as half of their product.
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Use the quadratic formula to solve. Express your answer in simplest form.
\newline
a
2
+
4
a
+
4
=
0
a^{2}+4 a+4=0
a
2
+
4
a
+
4
=
0
\newline
a
=
□
a=\square
a
=
□
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The function
f
f
f
is defined by
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x)=a x^{2}+b x+c
f
(
x
)
=
a
x
2
+
b
x
+
c
, where
a
,
b
a, b
a
,
b
, and
c
c
c
are constants and
1
<
a
<
4
1<a<4
1
<
a
<
4
. The graph of
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
in the
x
y
x y
x
y
-plane passes through points.
(
11
,
0
)
(11,0)
(
11
,
0
)
and
(
−
2
,
0
)
(-2,0)
(
−
2
,
0
)
. If
a
a
a
is an integer, what could be the value of
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x)=a x^{2}+b x+c
f
(
x
)
=
a
x
2
+
b
x
+
c
0
0
0
?
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Let
A
=
∫
0
1
cos
x
d
x
A=\int_{0}^{1}\cos x\,dx
A
=
∫
0
1
cos
x
d
x
. We estimate
A
A
A
using the
L
L
L
,
R
R
R
, and
\newline
T
T
T
approximations with
n
=
100
n=100
n
=
100
subintervals. Which is true?
\newline
(A)
L
<
A
<
T
<
R
L < A < T < R
L
<
A
<
T
<
R
\newline
(B)
L
<
T
<
A
<
R
L < T < A < R
L
<
T
<
A
<
R
\newline
(C)
R
<
A
<
T
<
L
R < A < T < L
R
<
A
<
T
<
L
\newline
(D)
R
<
T
<
A
<
L
R < T < A < L
R
<
T
<
A
<
L
Get tutor help
Let
g
(
x
)
=
x
−
5
x
−
4
−
1
g(x)=\frac{x-5}{\sqrt{x-4}-1}
g
(
x
)
=
x
−
4
−
1
x
−
5
when
x
≠
5
x \neq 5
x
=
5
.
\newline
g
g
g
is continuous for all
x
>
4
x>4
x
>
4
.
\newline
Find
g
(
5
)
g(5)
g
(
5
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
2
2
2
\newline
(B)
8
8
8
\newline
(C)
10
10
10
\newline
(D)
5
5
5
Get tutor help
−
6
i
-6 i
−
6
i
is a root of
f
(
x
)
=
x
3
−
20
x
2
+
f(x)=x^{3}-20 x^{2}+
f
(
x
)
=
x
3
−
20
x
2
+
36
x
−
720
36 x-720
36
x
−
720
. Find the other roots of
f
(
x
)
f(x)
f
(
x
)
.
\newline
Write your answer as a list of simplified values separated by commas, if there is more than one value.
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−
173
+
149
i
-173+149 i
−
173
+
149
i
is a root of
f
(
x
)
=
x
2
+
f(x)=x^{2}+
f
(
x
)
=
x
2
+
346
x
+
52130
346 x+52130
346
x
+
52130
. Find the other roots of
f
(
x
)
\mathrm{f}(x)
f
(
x
)
.
\newline
Write your answer as a list of simplified values separated by commas, if there is more than one value.
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Point
H
H
H
is located at
(
2
,
−
7
)
(2,-7)
(
2
,
−
7
)
. Select all of the following that are
7
7
7
units from point
H
H
H
.
\newline
Choose all answers that apply:
\newline
A x-axis
\newline
B
(
−
5
,
−
7
)
(-5,-7)
(
−
5
,
−
7
)
\newline
C
(
−
7
,
7
)
(-7,7)
(
−
7
,
7
)
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Determine the critical numbers, if any, of the function
f
f
f
on the interval
[
1
,
8
]
[1,8]
[
1
,
8
]
.
\newline
f
(
x
)
=
x
2
8
−
x
f(x)=x^{2}\sqrt{8-x}
f
(
x
)
=
x
2
8
−
x
\newline
Give your answer as a comma-separated list. Express numbers in exact form. If the function does not have any critical numbers, enter DNE.
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Find all critical points of the function
\newline
f
(
t
)
=
t
−
8
t
+
3
.
f(t)=t-8\sqrt{t+3}.
f
(
t
)
=
t
−
8
t
+
3
.
\newline
(Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.)
\newline
critical points:
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Find all critical points of the function
\newline
f
(
x
)
=
5
x
2
5
x
2
−
2
x
+
10
.
f(x)=\frac{5x^{2}}{5x^{2}-2x+10}.
f
(
x
)
=
5
x
2
−
2
x
+
10
5
x
2
.
\newline
(Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.)
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Find all critical points of the function
\newline
f
(
x
)
=
x
−
14
−
x
−
15
.
f(x)=x^{-14}-x^{-15}.
f
(
x
)
=
x
−
14
−
x
−
15
.
\newline
(Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list. If the function does not have any critical points, enter DNE.)
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Find all
x
x
x
-intercepts of the following function. Write your answer or answers as coordinate points. Be sure to select the appropriate number of
x
x
x
-intercepts.
\newline
f
(
x
)
=
2
x
−
4
5
x
−
8
f(x)=\frac{2 x-4}{5 x-8}
f
(
x
)
=
5
x
−
8
2
x
−
4
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{
−
x
+
3
=
y
−
6
x
+
18
=
6
y
\begin{cases} -x+3=y \ -6x+18=6y \end{cases}
{
−
x
+
3
=
y
−
6
x
+
18
=
6
y
\newline
Which of the following accurately describes all solutions to the system of equations shown?
\newline
Choose
1
1
1
answer:
\newline
(A)
x
=
0
x=0
x
=
0
and
y
=
3
y=3
y
=
3
\newline
(B)
x
=
3
x=3
x
=
3
and
y
=
0
y=0
y
=
0
\newline
(C) There are infinite solutions to the system.
\newline
(D) There are no solutions to the system.
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Solve the equation. Check your solution.
\newline
19
=
2
−
(
z
+
5
)
19=2-(z+5)
19
=
2
−
(
z
+
5
)
\newline
The solution set is
_
_
_
_
_
\_\_\_\_\_
_____
(Type an integer or a simplified fraction).
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Solve the equation. Check your solution.
\newline
−
3
(
4
y
+
2
)
−
12
=
−
8
(
y
+
4
)
+
4
y
-3(4 y+2)-12=-8(y+4)+4 y
−
3
(
4
y
+
2
)
−
12
=
−
8
(
y
+
4
)
+
4
y
\newline
The solution set is
_
_
_
_
_
\_\_\_\_\_
_____
. Type an integer or a simplified fraction.
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0.5
(
8
w
+
2
v
)
=
3
0.5(8w+2v)=3
0.5
(
8
w
+
2
v
)
=
3
\newline
8
w
=
2
−
v
+
4
w
8w=2-v+4w
8
w
=
2
−
v
+
4
w
\newline
Which of the following accurately describes all solutions to the system of equations shown?
\newline
Choose
1
1
1
answer:
\newline
(A)
v
=
1
v=1
v
=
1
and
w
=
1
4
w=\frac{1}{4}
w
=
4
1
\newline
(B)
v
=
4
v=4
v
=
4
and
w
=
−
1
4
w=-\frac{1}{4}
w
=
−
4
1
\newline
(C) There are infinite solutions to the system.
\newline
(D) There are no solutions to the system.
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4
n
−
30
=
2
(
2
n
+
15
)
4n-30=2(2n+15)
4
n
−
30
=
2
(
2
n
+
15
)
\newline
Which of the following best describes the solutions to the equation shown?
\newline
Choose
1
1
1
answer:
\newline
(A) There is exactly one solution,
n
=
0
n=0
n
=
0
.
\newline
(B) There is exactly one solution,
n
=
−
15
4
n=-\frac{15}{4}
n
=
−
4
15
.
\newline
(C) There are no solutions.
\newline
(D) There are infinitely many solutions.
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Solve for
u
u
u
and simplify your answer.
\newline
5
2
u
=
−
15
\frac{5}{2} u=-15
2
5
u
=
−
15
\newline
Answer:
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Solve by substitution
\newline
{
3
x
−
2
y
=
14
y
=
5
x
\begin{cases} 3x-2y=14 \ y=5x \end{cases}
{
3
x
−
2
y
=
14
y
=
5
x
\newline
A.
(
−
2
,
−
10
)
(-2,-10)
(
−
2
,
−
10
)
\newline
B.
(
−
2
,
10
)
(-2,10)
(
−
2
,
10
)
\newline
C.
(
2
,
−
10
)
(2,-10)
(
2
,
−
10
)
\newline
D. infinitely many solutions
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Convert
8
3
5
8 \frac{3}{5}
8
5
3
into an improper fraction.
\newline
Answer:
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Solve the equation for the given variable. If your answer is a fraction, write it in reduced, fractional form. Do NOT convert the answer to a decimal.
\newline
w
−
1
5
−
8
=
w
+
1
2
\frac{w-1}{5}-8=\frac{w+1}{2}
5
w
−
1
−
8
=
2
w
+
1
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Solve the given equation. (Enter your answers as a comma-separated list. Let
k
k
k
be any integer.
\newline
sin
(
θ
)
=
−
1
2
\sin(\theta) = -\frac{1}{2}
sin
(
θ
)
=
−
2
1
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4
sin
(
2
θ
)
−
7
sin
(
θ
)
=
0
4\sin(2\theta)-7\sin(\theta)=0
4
sin
(
2
θ
)
−
7
sin
(
θ
)
=
0
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Let
h
(
x
)
=
2
x
3
+
3
x
2
−
12
x
+
5
.
h(x)=2 x^{3}+3 x^{2}-12 x+5 \text {. }
h
(
x
)
=
2
x
3
+
3
x
2
−
12
x
+
5
.
\newline
The absolute minimum value of
h
h
h
over the closed interval
−
3
≤
x
≤
2
-3 \leq x \leq 2
−
3
≤
x
≤
2
occurs at what
x
x
x
value?
\newline
Choose
1
1
1
answer:
\newline
(A)
1
1
1
\newline
(B)
2
2
2
\newline
(C)
−
2
-2
−
2
\newline
(D)
−
3
-3
−
3
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1
2
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