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Math Problems
Algebra 2
Complex conjugate theorem
Find all solutions of the equation below.
\newline
5
3
x
−
2
=
2
5^{3 x-2}=2
5
3
x
−
2
=
2
\newline
The solution(s) is/are
x
=
□
x=\square
x
=
□
\newline
(Simplify your answer. Use a comma to separate answers as needed. Round to four decimal places as needed.)
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Select all the expressions that are equivalent to
(
2
1
)
2
(2^1)^2
(
2
1
)
2
.
\newline
Multi-select Choices:
\newline
(A)
2
3
2^3
2
3
\newline
(B)
1
2
2
\frac{1}{2^2}
2
2
1
\newline
(C)
2
2
2^2
2
2
\newline
(D)
1
2
3
\frac{1}{2^3}
2
3
1
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101
101
101
is a root of
f
(
x
)
=
x
2
+
10
,
201
f(x) = x^2 + 10,201
f
(
x
)
=
x
2
+
10
,
201
. 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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−
176
-176
−
176
is a root of
f
(
x
)
=
x
2
+
30
,
976
f(x) = x^2 + 30,976
f
(
x
)
=
x
2
+
30
,
976
. 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.
\newline
______
\newline
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In
△
V
W
X
,
x
=
2.1
c
m
,
w
=
9.5
c
m
\triangle \mathrm{VWX}, x=2.1 \mathrm{~cm}, w=9.5 \mathrm{~cm}
△
VWX
,
x
=
2.1
cm
,
w
=
9.5
cm
and
∠
W
=
11
0
∘
\angle \mathrm{W}=110^{\circ}
∠
W
=
11
0
∘
. Find all possible values of
∠
X
\angle \mathrm{X}
∠
X
, to the nearest
10
10
10
th of a degree.
\newline
Answer:
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In
△
Q
R
S
,
r
=
86
c
m
,
q
=
66
c
m
\triangle \mathrm{QRS}, r=86 \mathrm{~cm}, q=66 \mathrm{~cm}
△
QRS
,
r
=
86
cm
,
q
=
66
cm
and
∠
Q
=
5
∘
\angle \mathrm{Q}=5^{\circ}
∠
Q
=
5
∘
. Find all possible values of
∠
R
\angle \mathrm{R}
∠
R
, to the nearest degree.
\newline
Answer:
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Find all points
c
c
c
satisfying the conclusion of the MVT for the function
f
(
x
)
=
9
x
ln
(
x
)
+
9
f(x)=9 x \ln (x)+9
f
(
x
)
=
9
x
ln
(
x
)
+
9
and interval
[
1
,
9
]
[1,9]
[
1
,
9
]
. (Use decimal notation. Give your answer to three decimal places. Give your answer as comma-separated list.)
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Given that
1
−
i
1-i
1
−
i
is a zero of
p
(
x
)
=
x
3
−
6
x
2
+
10
x
−
8
p(x)=x^{3}-6x^{2}+10x-8
p
(
x
)
=
x
3
−
6
x
2
+
10
x
−
8
, find the remaining zeroes.
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Find all critical points of the function
\newline
f
(
x
)
=
9
x
+
∣
8
x
+
1
∣
f(x)=9x+|8x+1|
f
(
x
)
=
9
x
+
∣8
x
+
1∣
\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.)
\newline
critical points:
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Rewrite the expression as a product of four linear factors:
\newline
(
x
2
−
9
x
)
2
−
2
(
x
2
−
9
x
)
−
80
\left(x^{2}-9 x\right)^{2}-2\left(x^{2}-9 x\right)-80
(
x
2
−
9
x
)
2
−
2
(
x
2
−
9
x
)
−
80
\newline
Answer:
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Rewrite the expression as a product of four linear factors:
\newline
(
x
2
+
5
x
)
2
−
20
(
x
2
+
5
x
)
+
84
\left(x^{2}+5 x\right)^{2}-20\left(x^{2}+5 x\right)+84
(
x
2
+
5
x
)
2
−
20
(
x
2
+
5
x
)
+
84
\newline
Answer:
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Rewrite the expression as a product of four linear factors:
\newline
(
x
2
−
6
x
)
2
−
11
(
x
2
−
6
x
)
−
80
\left(x^{2}-6 x\right)^{2}-11\left(x^{2}-6 x\right)-80
(
x
2
−
6
x
)
2
−
11
(
x
2
−
6
x
)
−
80
\newline
Answer:
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Rewrite the expression as a product of four linear factors:
\newline
(
x
2
+
6
x
)
2
−
11
(
x
2
+
6
x
)
−
80
\left(x^{2}+6 x\right)^{2}-11\left(x^{2}+6 x\right)-80
(
x
2
+
6
x
)
2
−
11
(
x
2
+
6
x
)
−
80
\newline
Answer:
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Rewrite the expression as a product of four linear factors:
\newline
(
12
x
2
−
19
x
)
2
−
5
(
12
x
2
−
19
x
)
−
50
\left(12 x^{2}-19 x\right)^{2}-5\left(12 x^{2}-19 x\right)-50
(
12
x
2
−
19
x
)
2
−
5
(
12
x
2
−
19
x
)
−
50
\newline
Answer:
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Find the
z
z
z
-scores for which
15
%
15 \%
15%
of the distribution's area lies between
−
z
-z
−
z
and
z
z
z
.
\newline
The
z
z
z
-scores are
□
\square
□
.
\newline
(Use a comma to separate answers as needed. Round to two decimal places as needed.)
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Find the
1
3
th
13^{\text {th }}
1
3
th
term of the geometric sequence
7
,
−
21
,
63
,
…
7,-21,63, \ldots
7
,
−
21
,
63
,
…
\newline
Answer:
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Find all critical points of the function
\newline
f
(
x
)
=
3
x
2
7
x
2
−
2
x
+
2
.
f(x)=\frac{3x^{2}}{7x^{2}-2x+2}.
f
(
x
)
=
7
x
2
−
2
x
+
2
3
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.)
\newline
critical points:
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For the function
f
(
x
)
=
x
2
+
2
x
+
4
f(x)=x^{2}+2 x+4
f
(
x
)
=
x
2
+
2
x
+
4
, find the slope of the secant line between
x
=
1
x=1
x
=
1
and
x
=
4
x=4
x
=
4
.
\newline
Answer:
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For the function
f
(
x
)
=
x
2
+
2
x
−
5
f(x)=x^{2}+2 x-5
f
(
x
)
=
x
2
+
2
x
−
5
, find the slope of the secant line between
x
=
−
3
x=-3
x
=
−
3
and
x
=
5
x=5
x
=
5
.
\newline
Answer:
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62
+
58
i
62+58i
62
+
58
i
is a root of
f
(
x
)
=
x
2
–
124
x
+
7208
f(x) = x^2–124x+7208
f
(
x
)
=
x
2
–124
x
+
7208
. Find the other roots of
f
(
x
)
f(x)
f
(
x
)
.
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Let
f
(
x
)
=
−
18
x
f(x)=-\frac{18}{x}
f
(
x
)
=
−
x
18
and
g
(
x
)
=
x
2
+
10
x
+
27
.
g(x)=x^{2}+10 x+27 \text {. }
g
(
x
)
=
x
2
+
10
x
+
27
.
\newline
Find the sum of the areas enclosed by the graphs of
f
f
f
and
g
g
g
between
x
=
−
6
x=-6
x
=
−
6
and
x
=
−
1
x=-1
x
=
−
1
.
\newline
Use a graphing calculator and round your answer to three decimal places.
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Let
f
(
x
)
=
8
−
2
x
f(x)=8-2 x
f
(
x
)
=
8
−
2
x
and
g
(
x
)
=
x
3
−
7
x
2
+
12
x
.
g(x)=x^{3}-7 x^{2}+12 x \text {. }
g
(
x
)
=
x
3
−
7
x
2
+
12
x
.
\newline
Find the sum of the areas enclosed by the graphs of
f
f
f
and
g
g
g
between
x
=
1
x=1
x
=
1
and
x
=
4
x=4
x
=
4
.
\newline
Use a graphing calculator and round your answer to three decimal places.
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Let
f
(
x
)
=
1
2
x
3
+
1
2
x
2
−
x
f(x)=\frac{1}{2} x^{3}+\frac{1}{2} x^{2}-x
f
(
x
)
=
2
1
x
3
+
2
1
x
2
−
x
and
g
(
x
)
=
x
2
+
2
x
g(x)=x^{2}+2 x
g
(
x
)
=
x
2
+
2
x
.
\newline
Find the sum of the areas enclosed by the graphs of
f
f
f
and
g
g
g
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
3
x=3
x
=
3
.
\newline
Use a graphing calculator and round your answer to three decimal places.
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Let
f
(
x
)
=
1
2
x
3
−
2
x
+
3
f(x)=\frac{1}{2} x^{3}-2 x+3
f
(
x
)
=
2
1
x
3
−
2
x
+
3
and
g
(
x
)
=
x
2
+
7
2
x
−
3
g(x)=x^{2}+\frac{7}{2} x-3
g
(
x
)
=
x
2
+
2
7
x
−
3
\newline
Find the sum of the areas enclosed by the graphs of
f
f
f
and
g
g
g
between
x
=
−
3
x=-3
x
=
−
3
and
x
=
4
x=4
x
=
4
.
\newline
Use a graphing calculator and round your answer to three decimal places.
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A polynomial function
f
f
f
is defined as
f
(
x
)
=
3
(
x
−
4
)
(
2
x
−
1
)
f(x)=3(x-4)(2 x-1)
f
(
x
)
=
3
(
x
−
4
)
(
2
x
−
1
)
. What is the product of all of the zeros of function
f
f
f
?
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−
70
i
-70i
−
70
i
is a root of
f
(
x
)
=
x
2
+
4
,
900
f(x) = x^2 + 4,900
f
(
x
)
=
x
2
+
4
,
900
. 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.
\newline
______
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