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Math Problems
Algebra 2
Composition of linear and quadratic functions: find an equation
Given the function
f
(
x
)
=
−
x
2
+
1
2
f(x)=-x^{2}+\frac{1}{2}
f
(
x
)
=
−
x
2
+
2
1
, then what is
f
(
x
−
1
)
f(x-1)
f
(
x
−
1
)
as a simplified polynomial?
\newline
Answer:
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Given the function
f
(
x
)
=
−
1
3
x
2
−
1
2
x
f(x)=-\frac{1}{3} x^{2}-\frac{1}{2} x
f
(
x
)
=
−
3
1
x
2
−
2
1
x
, then what is
f
(
−
3
x
)
f(-3 x)
f
(
−
3
x
)
as a simplified polynomial?
\newline
Answer:
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Given the function
f
(
x
)
=
−
x
2
+
1
4
f(x)=-x^{2}+\frac{1}{4}
f
(
x
)
=
−
x
2
+
4
1
, then what is
f
(
x
+
2
)
f(x+2)
f
(
x
+
2
)
as a simplified polynomial?
\newline
Answer:
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Given the function
f
(
x
)
=
−
2
x
−
1
4
x
2
f(x)=-2 x-\frac{1}{4} x^{2}
f
(
x
)
=
−
2
x
−
4
1
x
2
, then what is
f
(
x
−
2
)
f(x-2)
f
(
x
−
2
)
as a simplified polynomial?
\newline
Answer:
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Given the function
f
(
x
)
=
5
x
3
+
2
5
x
f(x)=5 x^{3}+\frac{2}{5} x
f
(
x
)
=
5
x
3
+
5
2
x
, then what is
f
(
x
)
+
2
f(x)+2
f
(
x
)
+
2
as a simplified polynomial?
\newline
Answer:
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Given the function
f
(
x
)
=
−
3
2
−
x
2
f(x)=-\frac{3}{2}-x^{2}
f
(
x
)
=
−
2
3
−
x
2
, then what is
f
(
x
+
2
)
f(x+2)
f
(
x
+
2
)
as a simplified polynomial?
\newline
Answer:
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Given the function
f
(
x
)
=
−
4
3
x
2
−
2
3
f(x)=-\frac{4}{3} x^{2}-\frac{2}{3}
f
(
x
)
=
−
3
4
x
2
−
3
2
, then what is
f
(
x
)
−
3
f(x)-3
f
(
x
)
−
3
as a simplified polynomial?
\newline
Answer:
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Given the function
f
(
x
)
=
−
1
6
−
4
x
2
f(x)=-\frac{1}{6}-4 x^{2}
f
(
x
)
=
−
6
1
−
4
x
2
, then what is
f
(
3
x
)
f(3 x)
f
(
3
x
)
as a simplified polynomial?
\newline
Answer:
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Given the function
f
(
x
)
=
−
x
−
4
5
x
2
f(x)=-x-\frac{4}{5} x^{2}
f
(
x
)
=
−
x
−
5
4
x
2
, then what is
f
(
x
−
3
)
f(x-3)
f
(
x
−
3
)
as a simplified polynomial?
\newline
Answer:
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37
37
37
.
y
=
x
a
x
⇒
a
x
−
1
=
y=x a^{x} \Rightarrow a^{x-1}=
y
=
x
a
x
⇒
a
x
−
1
=
?
\newline
A)
y
x
−
1
\frac{y}{x-1}
x
−
1
y
\newline
B)
1
x
(
a
−
1
)
\frac{1}{x(a-1)}
x
(
a
−
1
)
1
\newline
C)
y
−
1
x
−
1
\frac{y-1}{x-1}
x
−
1
y
−
1
\newline
D)
y
a
x
\frac{y}{a x}
a
x
y
\newline
E)
y
−
x
x
\frac{y-x}{x}
x
y
−
x
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It is given that
10
x
3
+
5
x
2
−
x
+
4
=
(
2
x
−
1
)
(
x
+
3
)
Q
(
x
)
+
A
x
+
B
10 x^{3}+5 x^{2}-x+4=(2 x-1)(x+3) \mathrm{Q}(x)+A x+B
10
x
3
+
5
x
2
−
x
+
4
=
(
2
x
−
1
)
(
x
+
3
)
Q
(
x
)
+
A
x
+
B
, where
Q
(
x
)
\mathrm{Q}(x)
Q
(
x
)
is a polynomial. Find the values of the constants
A
A
A
and
B
B
B
.
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For the function,
f
(
x
)
=
x
2
−
3
x
+
9
f(x)=x^{2}-3 x+9
f
(
x
)
=
x
2
−
3
x
+
9
\newline
Find when
f
(
x
)
=
7
f(x)=7
f
(
x
)
=
7
.
\newline
x
=
□
x= \square
x
=
□
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What is the value of the expression below when
y
=
4
y=4
y
=
4
?
\newline
y
2
+
7
y
+
9
y^{2}+7 y+9
y
2
+
7
y
+
9
\newline
Answer:
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What is the value of the expression below when
z
=
5
z=5
z
=
5
?
\newline
z
2
−
z
+
6
z^{2}-z+6
z
2
−
z
+
6
\newline
Answer:
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Solve the equation by factoring:
\newline
x
3
−
3
x
2
−
10
x
=
0
x^{3}-3 x^{2}-10 x=0
x
3
−
3
x
2
−
10
x
=
0
\newline
Answer:
x
=
x=
x
=
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Simplify the expression completely if possible.
\newline
x
2
+
13
x
+
40
x
2
+
9
x
+
8
\frac{x^{2}+13 x+40}{x^{2}+9 x+8}
x
2
+
9
x
+
8
x
2
+
13
x
+
40
\newline
Answer:
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Use the quadratic formula to solve. Express your answer in simplest form.
\newline
8
w
2
−
27
w
+
25
=
3
w
8 w^{2}-27 w+25=3 w
8
w
2
−
27
w
+
25
=
3
w
\newline
Answer:
w
=
w=
w
=
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Use the quadratic formula to solve. Express your answer in simplest form.
\newline
8
t
2
−
19
t
+
12
=
4
t
2
8 t^{2}-19 t+12=4 t^{2}
8
t
2
−
19
t
+
12
=
4
t
2
\newline
Answer:
t
=
t=
t
=
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Use the quadratic formula to solve. Express your answer in simplest form.
\newline
−
2
w
2
+
11
w
+
5
=
−
4
w
2
-2 w^{2}+11 w+5=-4 w^{2}
−
2
w
2
+
11
w
+
5
=
−
4
w
2
\newline
Answer:
w
=
w=
w
=
Get tutor help
Given
f
(
x
)
=
−
x
2
+
3
x
+
9
f(x)=-x^{2}+3 x+9
f
(
x
)
=
−
x
2
+
3
x
+
9
, find
f
(
10
)
f(10)
f
(
10
)
\newline
Answer:
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Given
f
(
x
)
=
−
3
x
2
−
10
x
−
1
f(x)=-3 x^{2}-10 x-1
f
(
x
)
=
−
3
x
2
−
10
x
−
1
, find
f
(
−
7
)
f(-7)
f
(
−
7
)
\newline
Answer:
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Solve for all values of
x
x
x
:
\newline
(
6
x
+
5
)
2
+
(
6
x
+
5
)
=
0
(6 x+5)^{2}+(6 x+5)=0
(
6
x
+
5
)
2
+
(
6
x
+
5
)
=
0
\newline
Answer:
x
=
x=
x
=
Get tutor help
Solve for all values of
x
x
x
:
\newline
7
(
4
x
+
3
)
−
(
4
x
+
3
)
2
=
0
7(4 x+3)-(4 x+3)^{2}=0
7
(
4
x
+
3
)
−
(
4
x
+
3
)
2
=
0
\newline
Answer:
x
=
x=
x
=
Get tutor help
Solve for all values of
x
x
x
:
\newline
(
x
−
7
)
2
−
10
(
x
−
7
)
=
0
(x-7)^{2}-10(x-7)=0
(
x
−
7
)
2
−
10
(
x
−
7
)
=
0
\newline
Answer:
x
=
x=
x
=
Get tutor help
Find the
6
5
th
65^{\text {th }}
6
5
th
term of the arithmetic sequence
−
26
,
−
11
,
4
,
…
-26,-11,4, \ldots
−
26
,
−
11
,
4
,
…
\newline
Answer:
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Factor the expression completely.
\newline
x
2
y
3
−
x
y
4
x^{2} y^{3}-x y^{4}
x
2
y
3
−
x
y
4
\newline
Answer:
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Factor the expression completely.
\newline
x
4
y
+
x
2
y
x^{4} y+x^{2} y
x
4
y
+
x
2
y
\newline
Answer:
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Factor the expression completely.
\newline
x
y
3
+
x
3
y
x y^{3}+x^{3} y
x
y
3
+
x
3
y
\newline
Answer:
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Factor the expression completely.
\newline
x
2
y
2
+
x
y
x^{2} y^{2}+x y
x
2
y
2
+
x
y
\newline
Answer:
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Factor the expression completely.
\newline
x
4
−
3
x
2
+
2
x^{4}-3 x^{2}+2
x
4
−
3
x
2
+
2
\newline
Answer:
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Factor the expression completely.
\newline
x
4
−
3
x
2
−
4
x^{4}-3 x^{2}-4
x
4
−
3
x
2
−
4
\newline
Answer:
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Factor the expression completely.
\newline
x
4
−
4
x
2
+
3
x^{4}-4 x^{2}+3
x
4
−
4
x
2
+
3
\newline
Answer:
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Factor completely.
\newline
2
x
2
+
32
x
+
126
2 x^{2}+32 x+126
2
x
2
+
32
x
+
126
\newline
Answer:
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How many solutions are there to the following system of equations:
\newline
y
=
6
−
x
y=6-x
y
=
6
−
x
\newline
y
=
x
2
−
6
x
+
6
y=x^{2}-6x+6
y
=
x
2
−
6
x
+
6
Get tutor help
For the following equation, evaluate
f
′
(
1
)
f^{\prime}(1)
f
′
(
1
)
.
\newline
f
(
x
)
=
2
x
5
+
x
2
−
3
f(x)=2 x^{5}+x^{2}-3
f
(
x
)
=
2
x
5
+
x
2
−
3
\newline
Answer:
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For the following equation, find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
.
\newline
f
(
x
)
=
4
x
2
−
3
x
−
9
f(x)=4 x^{2}-3 x-9
f
(
x
)
=
4
x
2
−
3
x
−
9
\newline
Answer:
f
′
(
x
)
=
f^{\prime}(x)=
f
′
(
x
)
=
Get tutor help
Write the expression as a single power of
m
m
m
.
\newline
m
3
m
\frac{m^{3}}{m}
m
m
3
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Given the function
f
(
x
)
=
2
x
2
+
3
3
+
5
x
2
f(x)=\frac{2 x^{2}+3}{3+5 x^{2}}
f
(
x
)
=
3
+
5
x
2
2
x
2
+
3
, find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
in simplified form.
\newline
Answer:
f
′
(
x
)
=
f^{\prime}(x)=
f
′
(
x
)
=
Get tutor help
Given the function
f
(
x
)
=
(
−
1
−
10
x
+
5
x
−
2
)
(
6
x
−
2
−
9
)
f(x)=\left(-1-10 x+5 x^{-2}\right)\left(6 x^{-2}-9\right)
f
(
x
)
=
(
−
1
−
10
x
+
5
x
−
2
)
(
6
x
−
2
−
9
)
, find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
in any form.
\newline
Answer:
f
′
(
x
)
=
f^{\prime}(x)=
f
′
(
x
)
=
Get tutor help
Given the function
f
(
x
)
=
(
−
9
−
5
x
3
)
(
7
x
+
2
x
−
3
+
2
)
f(x)=\left(-9-5 x^{3}\right)\left(7 x+2 x^{-3}+2\right)
f
(
x
)
=
(
−
9
−
5
x
3
)
(
7
x
+
2
x
−
3
+
2
)
, find
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
in any form.
\newline
Answer:
f
′
(
x
)
=
f^{\prime}(x)=
f
′
(
x
)
=
Get tutor help
d
y
d
t
=
y
\frac{d y}{d t}=y
d
t
d
y
=
y
, and
y
=
1
y=1
y
=
1
when
t
=
4
t=4
t
=
4
.
\newline
Solve the equation.
\newline
Choose
1
1
1
answer:
\newline
(A)
y
=
e
t
−
4
y=e^{t-4}
y
=
e
t
−
4
\newline
(B)
y
=
4
e
t
−
1
y=4 e^{t-1}
y
=
4
e
t
−
1
\newline
(C)
y
=
e
4
t
y=e^{4 t}
y
=
e
4
t
\newline
(D)
y
=
4
e
t
y=4 e^{t}
y
=
4
e
t
Get tutor help
1
2
−
60
%
−
25
%
=
\frac{1}{2}-60 \%-25 \%=
2
1
−
60%
−
25%
=
\newline
Enter the answer as an exact decimal or simplified fraction.
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−
2
i
⋅
(
2
+
6
i
)
=
-2 i \cdot(2+6 i)=
−
2
i
⋅
(
2
+
6
i
)
=
\newline
Your answer should be a complex number in the form
a
+
b
i
a+b i
a
+
bi
where
a
a
a
and
b
b
b
are real numbers.
Get tutor help
Divide the polynomials.
\newline
Your answer should be a polynomial.
\newline
3
x
5
−
x
x
=
□
\frac{3 x^{5}-x}{x}=\square
x
3
x
5
−
x
=
□
Get tutor help
Divide the polynomials.
\newline
Your answer should be a polynomial.
\newline
3
x
4
−
6
x
2
−
x
x
=
\frac{3 x^{4}-6 x^{2}-x}{x}=
x
3
x
4
−
6
x
2
−
x
=
Get tutor help
Divide the polynomials.
\newline
Your answer should be a polynomial.
\newline
2
x
3
−
5
x
2
+
x
x
=
\frac{2 x^{3}-5 x^{2}+x}{x}=
x
2
x
3
−
5
x
2
+
x
=
Get tutor help
Divide the polynomials.
\newline
Your answer should be a polynomial.
\newline
2
x
4
−
3
x
x
=
\frac{2 x^{4}-3 x}{x}=
x
2
x
4
−
3
x
=
Get tutor help
Divide the polynomials.
\newline
The form of your answer should either be
p
(
x
)
p(x)
p
(
x
)
or
p
(
x
)
+
k
x
−
2
p(x)+\frac{k}{x-2}
p
(
x
)
+
x
−
2
k
where
p
(
x
)
p(x)
p
(
x
)
is a polynomial and
k
k
k
is an integer.
\newline
x
3
+
6
x
2
−
5
x
x
−
2
=
\frac{x^{3}+6 x^{2}-5 x}{x-2}=
x
−
2
x
3
+
6
x
2
−
5
x
=
Get tutor help
Divide the polynomials. The form of your answer should either be
p
(
x
)
p(x)
p
(
x
)
or
p
(
x
)
+
k
x
+
1
p(x)+\frac{k}{x+1}
p
(
x
)
+
x
+
1
k
where
p
(
x
)
p(x)
p
(
x
)
is a polynomial and
k
k
k
is an integer.
\newline
3
x
3
+
10
x
2
+
7
x
x
+
1
=
\frac{3 x^{3}+10 x^{2}+7 x}{x+1}=
x
+
1
3
x
3
+
10
x
2
+
7
x
=
Get tutor help
Divide the polynomials.
\newline
The form of your answer should either be
p
(
x
)
p(x)
p
(
x
)
or
p
(
x
)
+
k
x
−
4
p(x)+\frac{k}{x-4}
p
(
x
)
+
x
−
4
k
where
p
(
x
)
p(x)
p
(
x
)
is a polynomial and
k
k
k
is an integer.
\newline
4
x
3
−
14
x
2
−
7
x
−
4
x
−
4
=
\frac{4 x^{3}-14 x^{2}-7 x-4}{x-4}=
x
−
4
4
x
3
−
14
x
2
−
7
x
−
4
=
Get tutor help
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