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Math Problems
Calculus
Find derivatives of using multiple formulae
3
a
−
4
+
2
a
+
3
−
21
a
2
−
a
−
12
\frac{3}{a-4}+\frac{2}{a+3}-\frac{21}{a^{2}-a-12}
a
−
4
3
+
a
+
3
2
−
a
2
−
a
−
12
21
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(
4
x
2
−
10
x
)
+
(
2
x
2
+
5
x
−
5
)
\left(4 x^{2}-10 x\right)+\left(2 x^{2}+5 x-5\right)
(
4
x
2
−
10
x
)
+
(
2
x
2
+
5
x
−
5
)
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Simplify the expression :
4
a
−
3
b
3
⋅
2
a
2
b
−
1
⋅
4
a
2
b
2
4 a^{-3} b^{3} \cdot 2 a^{2} b^{-1} \cdot 4 a^{2} b^{2}
4
a
−
3
b
3
⋅
2
a
2
b
−
1
⋅
4
a
2
b
2
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x
2
−
2
x
+
y
2
−
4
y
−
4
=
0
x^{2}-2 x+y^{2}-4 y-4=0
x
2
−
2
x
+
y
2
−
4
y
−
4
=
0
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16
x
3
−
8
x
2
−
4
x
4
x
\frac{16 x^{3}-8 x^{2}-4 x}{4 x}
4
x
16
x
3
−
8
x
2
−
4
x
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39
39
39
.
a
−
1
=
x
2
+
x
+
1
,
b
=
x
3
−
1
3
⇒
a
2
b
3
=
a^{-1}=\sqrt{x^{2}+x+1}, b=\sqrt[3]{x^{3}-1} \Rightarrow a^{2} b^{3}=
a
−
1
=
x
2
+
x
+
1
,
b
=
3
x
3
−
1
⇒
a
2
b
3
=
?
\newline
A)
x
−
1
x-1
x
−
1
\newline
B)
(
x
+
1
)
2
(
x
3
−
1
)
(x+1)^{2}\left(x^{3}-1\right)
(
x
+
1
)
2
(
x
3
−
1
)
\newline
C)
x
−
1
x
\frac{x-1}{x}
x
x
−
1
\newline
D)
x
2
−
1
x^{2}-1
x
2
−
1
\newline
E)
x
2
−
1
x
2
\frac{x^{2}-1}{x^{2}}
x
2
x
2
−
1
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Factorise each of the following expressions completely.
\newline
(a)
1
4
a
2
−
a
b
−
15
b
2
\frac{1}{4} a^{2}-a b-15 b^{2}
4
1
a
2
−
ab
−
15
b
2
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f
(
x
)
=
4
x
x
3
+
2
x
2
−
3
x
f(x)=\frac{4 x}{x^{3}+2 x^{2}-3 x}
f
(
x
)
=
x
3
+
2
x
2
−
3
x
4
x
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2
2
2
. Which expression below simplifies to
2
(
z
2
−
2
)
3
w
2
\frac{2\left(z^{2}-2\right)}{3 w^{2}}
3
w
2
2
(
z
2
−
2
)
?
\newline
A.
(
4
w
2
z
4
−
16
w
2
z
2
)
9
w
6
z
4
\frac{\left(4 w^{2} z^{4}-16 w^{2} z^{2}\right)}{9 w^{6} z^{4}}
9
w
6
z
4
(
4
w
2
z
4
−
16
w
2
z
2
)
\newline
B.
(
4
w
2
z
4
−
16
w
2
z
2
)
9
w
4
z
2
\frac{\left(4 w^{2} z^{4}-16 w^{2} z^{2}\right)}{9 w^{4} z^{2}}
9
w
4
z
2
(
4
w
2
z
4
−
16
w
2
z
2
)
\newline
C.
(
12
w
2
z
4
−
24
w
2
z
2
)
18
w
6
z
4
\frac{\left(12 w^{2} z^{4}-24 w^{2} z^{2}\right)}{18 w^{6} z^{4}}
18
w
6
z
4
(
12
w
2
z
4
−
24
w
2
z
2
)
\newline
(D.
(
12
w
2
z
4
−
24
w
2
z
2
)
18
w
4
z
2
\frac{\left(12 w^{2} z^{4}-24 w^{2} z^{2}\right)}{18 w^{4} z^{2}}
18
w
4
z
2
(
12
w
2
z
4
−
24
w
2
z
2
)
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4
x
2
−
12
x
y
+
9
y
2
4
x
2
−
9
y
2
\frac{\frac{4}{x^{2}} - \frac{12}{xy} + \frac{9}{y^{2}} }{\frac{4}{x^{2}} - \frac{9}{y^{2}}}
x
2
4
−
y
2
9
x
2
4
−
x
y
12
+
y
2
9
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Divide.
\newline
37
z
4
v
5
+
24
z
v
5
−
32
z
5
v
6
−
4
z
2
v
3
\frac{37 z^{4} v^{5}+24 z v^{5}-32 z^{5} v^{6}}{-4 z^{2} v^{3}}
−
4
z
2
v
3
37
z
4
v
5
+
24
z
v
5
−
32
z
5
v
6
\newline
Simplify your answer as much as possible.
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Simplify the rational expression.
\newline
2
x
2
−
5
x
−
3
6
x
3
+
3
x
2
+
4
x
+
2
\frac{2 x^{2}-5 x-3}{6 x^{3}+3 x^{2}+4 x+2}
6
x
3
+
3
x
2
+
4
x
+
2
2
x
2
−
5
x
−
3
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Simplify the rational expression.
\newline
2
x
2
−
5
x
−
12
x
2
−
7
x
+
12
\frac{2 x^{2}-5 x-12}{x^{2}-7 x+12}
x
2
−
7
x
+
12
2
x
2
−
5
x
−
12
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x
−
7
3
x
2
+
x
−
2
+
x
2
+
3
x
−
18
x
2
−
5
x
+
6
×
x
2
+
3
x
−
10
3
x
2
+
13
x
−
10
\frac{x-7}{3 x^{2}+x-2}+\frac{x^{2}+3 x-18}{x^{2}-5 x+6} \times \frac{x^{2}+3 x-10}{3 x^{2}+13 x-10}
3
x
2
+
x
−
2
x
−
7
+
x
2
−
5
x
+
6
x
2
+
3
x
−
18
×
3
x
2
+
13
x
−
10
x
2
+
3
x
−
10
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8
x
y
−
2
×
3
x
−
3
y
3
÷
12
x
y
8 x y^{-2} \times 3 x^{-3} y^{3} \div 12 x y
8
x
y
−
2
×
3
x
−
3
y
3
÷
12
x
y
Get tutor help
Expand and simplify
(
x
−
5
)
(
x
+
1
)
(x-5)(x+1)
(
x
−
5
)
(
x
+
1
)
\newline
(A)
x
2
+
6
x
+
5
x^{2}+6 x+5
x
2
+
6
x
+
5
\newline
(B)
x
2
−
4
x
−
5
x^{2}-4 x-5
x
2
−
4
x
−
5
\newline
(C)
x
2
−
6
x
+
5
x^{2}-6 x+5
x
2
−
6
x
+
5
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Prove
cot
5
4
∘
tan
3
6
∘
+
tan
2
0
∘
cot
7
0
∘
−
2
=
0
\frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}}-2=0
t
a
n
3
6
∘
c
o
t
5
4
∘
+
c
o
t
7
0
∘
t
a
n
2
0
∘
−
2
=
0
.
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x
2
+
(
1
−
a
)
x
+
a
2
−
6
4
a
−
3
4
=
0
x^{2}+(1-a) x+a^{2}-\frac{6}{4} a-\frac{3}{4}=0
x
2
+
(
1
−
a
)
x
+
a
2
−
4
6
a
−
4
3
=
0
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(
4
c
3
d
−
1
)
4
2
−
1
c
−
2
d
5
\frac{\left(4 c^{3} d^{-1}\right)^{4}}{2^{-1} c^{-2} d^{5}}
2
−
1
c
−
2
d
5
(
4
c
3
d
−
1
)
4
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Solve the following equations.
\newline
(a)
3
x
−
5
+
3
=
x
\sqrt{3 x-5}+3=x
3
x
−
5
+
3
=
x
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∫
1
3
3
x
2
−
2
x
−
2
d
x
2
\frac{\int_{1}^{3} 3 x^{2}-2 x-2 d x}{2}
2
∫
1
3
3
x
2
−
2
x
−
2
d
x
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Select all the expressions that are equivalent to
(
1
0
3
)
1
(10^3)^1
(
1
0
3
)
1
.
\newline
Multi-select Choices:
\newline
(A)
1
1
0
4
\frac{1}{10^4}
1
0
4
1
\newline
(B)
1
0
3
10^3
1
0
3
\newline
(C)
1
0
4
10^4
1
0
4
\newline
(D)
1
1
0
3
\frac{1}{10^3}
1
0
3
1
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Select all the expressions that are equivalent to
1
2
4
×
1
2
6
12^4 \times 12^6
1
2
4
×
1
2
6
.
\newline
Multi-select Choices:
\newline
(A)
1
1
2
10
\frac{1}{12^{10}}
1
2
10
1
\newline
(B)
1
2
10
12^{10}
1
2
10
\newline
(C)
1
1
2
24
\frac{1}{12^{24}}
1
2
24
1
\newline
(D)
1
2
24
12^{24}
1
2
24
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Select all the expressions that are equivalent to
5
−
6
×
5
−
3
5^{-6} \times 5^{-3}
5
−
6
×
5
−
3
.
\newline
Multi-select Choices:
\newline
(A)
5
−
9
5^{-9}
5
−
9
\newline
(B)
1
5
9
\frac{1}{5^{9}}
5
9
1
\newline
(C)
5
18
5^{18}
5
18
\newline
(D)
1
5
−
9
\frac{1}{5^{-9}}
5
−
9
1
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Select all the expressions that are equivalent to
5
5
×
5
−
3
5^5 \times 5^{-3}
5
5
×
5
−
3
.
\newline
Multi-select Choices:
\newline
(A)
1
5
−
15
\frac{1}{5^{-15}}
5
−
15
1
\newline
(B)
5
−
15
5^{-15}
5
−
15
\newline
(C)
1
5
−
2
\frac{1}{5^{-2}}
5
−
2
1
\newline
(D)
1
5
2
\frac{1}{5^2}
5
2
1
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Factor completely.
\newline
x
5
−
x
4
+
3
x
−
3
=
x^{5}-x^{4}+3 x-3=
x
5
−
x
4
+
3
x
−
3
=
\newline
□
\square
□
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4
x
3
+
72
x
2
=
−
324
x
4 x^{3}+72 x^{2}=-324 x
4
x
3
+
72
x
2
=
−
324
x
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lim
t
→
0
(
1
(
1
/
sin
2
t
)
+
2
(
1
/
sin
2
t
)
+
…
.
+
n
(
1
/
sin
2
t
)
)
sin
2
t
\lim_{t \rightarrow 0}(1^{(1/\sin^{2}t)}+2^{(1/\sin^{2}t)}+\dots.+n^{(1/\sin^{2}t)})^{\sin^{2}t}
lim
t
→
0
(
1
(
1/
s
i
n
2
t
)
+
2
(
1/
s
i
n
2
t
)
+
…
.
+
n
(
1/
s
i
n
2
t
)
)
s
i
n
2
t
is equal to
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When factored completely,
m
5
+
m
3
−
6
m
m^{5}+m^{3}-6 m
m
5
+
m
3
−
6
m
is equivalent to
\newline
(
1
1
1
)
(
m
+
3
)
(
m
−
2
)
(m+3)(m-2)
(
m
+
3
)
(
m
−
2
)
\newline
(
2
2
2
)
(
m
3
+
3
m
)
(
m
2
−
2
)
\left (m^{3}+3 m\right)\left(m^{2}-2\right)
(
m
3
+
3
m
)
(
m
2
−
2
)
\newline
(
3
3
3
)
m
(
m
4
+
m
2
−
6
)
m\left(m^{4}+m^{2}-6\right)
m
(
m
4
+
m
2
−
6
)
\newline
(
4
4
4
)
m
(
m
2
+
3
)
(
m
2
−
2
)
m\left(m^{2}+3\right)\left(m^{2}-2\right)
m
(
m
2
+
3
)
(
m
2
−
2
)
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(
m
5
n
3
p
−
4
m
−
5
n
−
2
)
3
\left(\frac{m^{5} n^{3} p^{-4}}{m^{-5} n^{-2}}\right)^{3}
(
m
−
5
n
−
2
m
5
n
3
p
−
4
)
3
\newline
Write your answer using only positive exponents.
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4
x
2
+
1
x
3
−
4
x
\frac{4 x^{2}+1}{x^{3}-4 x}
x
3
−
4
x
4
x
2
+
1
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lim
x
→
5
x
2
−
25
x
−
5
=
_
_
_
_
_
\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}=\_\_\_\_\_
x
→
5
lim
x
−
5
x
2
−
25
=
_____
\newline
(A)
7
7
7
\newline
(B)
5
5
5
\newline
(C)
10
10
10
\newline
(D)
9
9
9
\newline
(E)
0
0
0
\newline
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Perform the following operation and express in simplest form.
\newline
x
3
+
2
x
2
6
x
÷
x
2
+
9
x
+
14
x
2
−
49
\frac{x^{3}+2 x^{2}}{6 x} \div \frac{x^{2}+9 x+14}{x^{2}-49}
6
x
x
3
+
2
x
2
÷
x
2
−
49
x
2
+
9
x
+
14
\newline
Answer:
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Perform the following operation and express in simplest form.
\newline
x
+
5
x
2
+
4
x
−
5
÷
9
x
+
9
x
2
−
1
\frac{x+5}{x^{2}+4 x-5} \div \frac{9 x+9}{x^{2}-1}
x
2
+
4
x
−
5
x
+
5
÷
x
2
−
1
9
x
+
9
\newline
Answer:
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Perform the following operation and express in simplest form.
\newline
x
+
1
4
x
−
20
÷
x
2
−
1
x
2
+
x
−
2
\frac{x+1}{4 x-20} \div \frac{x^{2}-1}{x^{2}+x-2}
4
x
−
20
x
+
1
÷
x
2
+
x
−
2
x
2
−
1
\newline
Answer:
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Perform the following operation and express in simplest form.
\newline
x
2
+
3
x
x
2
−
5
x
−
24
÷
4
x
+
32
x
2
−
64
\frac{x^{2}+3 x}{x^{2}-5 x-24} \div \frac{4 x+32}{x^{2}-64}
x
2
−
5
x
−
24
x
2
+
3
x
÷
x
2
−
64
4
x
+
32
\newline
Answer:
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Perform the operation and simplify the answer fully.
\newline
x
3
2
÷
5
x
2
3
\frac{x^{3}}{2} \div \frac{5 x^{2}}{3}
2
x
3
÷
3
5
x
2
\newline
Answer:
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Perform the operation and simplify the answer fully.
\newline
x
2
2
÷
5
x
2
\frac{x^{2}}{2} \div \frac{5 x}{2}
2
x
2
÷
2
5
x
\newline
Answer:
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Factor the following quadratic expressions, if possible.
\newline
(a).
k
2
−
12
k
+
20
k^{2}-12k+20
k
2
−
12
k
+
20
\newline
(b).
6
x
2
+
17
x
−
14
6x^{2}+17x-14
6
x
2
+
17
x
−
14
\newline
(c).
x
2
−
8
x
+
16
x^{2}-8x+16
x
2
−
8
x
+
16
\newline
(d).
9
m
2
−
1
9m^{2}-1
9
m
2
−
1
\newline
(e). Parts
a
a
a
through
e
e
e
are trinomials while part
d
d
d
is a binomial, yet they are all quadratics. What makes each of them a quadratic?
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Calculate the limit.
\newline
lim
(
x
,
y
)
→
(
0
,
0
)
x
2
+
y
4
x
4
+
y
2
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{4}}{x^{4}+y^{2}}
lim
(
x
,
y
)
→
(
0
,
0
)
x
4
+
y
2
x
2
+
y
4
Get tutor help
Prove that
1
csc
2
θ
+
sec
2
θ
+
1
sec
2
θ
=
2
+
sec
2
θ
csc
2
θ
\frac{1}{\csc^{2}\theta}+\sec^{2}\theta+\frac{1}{\sec^{2}\theta}=2+\frac{\sec^{2}\theta}{\csc^{2}\theta}
c
s
c
2
θ
1
+
sec
2
θ
+
s
e
c
2
θ
1
=
2
+
c
s
c
2
θ
s
e
c
2
θ
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What is the simplified form of the expression
\newline
(
6
x
4
+
4
x
3
−
2
x
2
+
5
)
−
(
3
x
4
−
2
x
3
+
x
+
4
)
(6x^{4}+4x^{3}-2x^{2}+5)-(3x^{4}-2x^{3}+x+4)
(
6
x
4
+
4
x
3
−
2
x
2
+
5
)
−
(
3
x
4
−
2
x
3
+
x
+
4
)
?
\newline
A.
3
x
4
+
2
x
3
−
2
x
2
+
x
+
1
3x^{4}+2x^{3}-2x^{2}+x+1
3
x
4
+
2
x
3
−
2
x
2
+
x
+
1
\newline
B.
3
x
4
+
2
x
3
−
2
x
2
−
x
+
9
3x^{4}+2x^{3}-2x^{2}-x+9
3
x
4
+
2
x
3
−
2
x
2
−
x
+
9
\newline
C.
3
x
4
+
6
x
3
−
2
x
2
+
x
+
1
3x^{4}+6x^{3}-2x^{2}+x+1
3
x
4
+
6
x
3
−
2
x
2
+
x
+
1
\newline
D.
3
x
4
+
6
x
3
−
2
x
2
−
x
+
1
3x^{4}+6x^{3}-2x^{2}-x+1
3
x
4
+
6
x
3
−
2
x
2
−
x
+
1
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(
w
−
3
)
5
−
(
w
−
3
)
2
(w-3)^{5}-(w-3)^{2}
(
w
−
3
)
5
−
(
w
−
3
)
2
\newline
Which of the following is equivalent to the given expression?
\newline
Choose
1
1
1
answer:
\newline
(A)
(
w
−
3
)
3
(w-3)^{3}
(
w
−
3
)
3
\newline
(B)
w
5
−
w
2
−
252
w^{5}-w^{2}-252
w
5
−
w
2
−
252
\newline
(C)
(
w
−
3
)
2
(
w
−
3
)
3
(w-3)^{2}(w-3)^{3}
(
w
−
3
)
2
(
w
−
3
)
3
\newline
(D)
(
w
−
3
)
2
(
(
w
−
3
)
3
−
1
)
(w-3)^{2}((w-3)^{3}-1)
(
w
−
3
)
2
((
w
−
3
)
3
−
1
)
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Calculate the limit.
\newline
lim
x
→
4
∣
x
+
1
∣
∣
x
−
4
∣
x
2
−
5
x
+
4
\lim _{x \rightarrow 4} \frac{|x+1||x-4|}{x^{2}-5 x+4}
lim
x
→
4
x
2
−
5
x
+
4
∣
x
+
1∣∣
x
−
4∣
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Which expression is equivalent to
8
w
+
4
w
8w + 4w
8
w
+
4
w
?
\newline
Choices:
\newline
(A)
w
+
12
w + 12
w
+
12
\newline
(B)
12
w
12w
12
w
\newline
(C)
w
12
w^{12}
w
12
\newline
(D)
11
w
11w
11
w
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Simplify.
\newline
12
x
3
y
3
+
10
x
2
y
3
−
12
x
2
y
2
+
12
x
y
3
2
x
y
\frac{12 x^{3} y^{3}+10 x^{2} y^{3}-12 x^{2} y^{2}+12 x y^{3}}{2 x y}
2
x
y
12
x
3
y
3
+
10
x
2
y
3
−
12
x
2
y
2
+
12
x
y
3
Get tutor help
Simplify and combine Like terms :
(
4
x
2
y
2
+
3
x
2
y
−
2
x
y
2
+
4
x
y
)
−
(
−
3
x
2
y
2
+
3
x
2
y
+
2
x
y
2
−
2
x
y
)
\left(4 x^{2} y^{2}+3 x^{2} y-2 x y^{2}+4 x y\right)-\left(-3 x^{2} y^{2}+3 x^{2} y+2 x y^{2}-2 x y\right)
(
4
x
2
y
2
+
3
x
2
y
−
2
x
y
2
+
4
x
y
)
−
(
−
3
x
2
y
2
+
3
x
2
y
+
2
x
y
2
−
2
x
y
)
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If
α
,
β
,
γ
,
δ
∈
R
\alpha,\beta,\gamma,\delta \in \mathbb{R}
α
,
β
,
γ
,
δ
∈
R
satisfy
(
α
+
1
)
2
+
(
β
+
1
)
2
+
(
γ
+
1
)
2
+
(
δ
+
1
)
2
α
+
β
+
γ
+
δ
=
4
\frac{(\alpha+1)^{2}+(\beta+1)^{2}+(\gamma+1)^{2}+(\delta+1)^{2}}{\alpha+\beta+\gamma+\delta}=4
α
+
β
+
γ
+
δ
(
α
+
1
)
2
+
(
β
+
1
)
2
+
(
γ
+
1
)
2
+
(
δ
+
1
)
2
=
4
If biquadratic equation
a
0
x
4
+
a
1
x
3
+
a
2
x
2
+
a
3
x
+
a
4
=
0
a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{3}x+a_{4}=0
a
0
x
4
+
a
1
x
3
+
a
2
x
2
+
a
3
x
+
a
4
=
0
has the roots
(
α
+
1
β
−
1
)
,
(
β
+
1
γ
−
1
)
,
(
γ
+
1
δ
−
1
)
,
(
δ
+
1
α
−
1
)
(\alpha+\frac{1}{\beta}-1),(\beta+\frac{1}{\gamma}-1),(\gamma+\frac{1}{\delta}-1),(\delta+\frac{1}{\alpha}-1)
(
α
+
β
1
−
1
)
,
(
β
+
γ
1
−
1
)
,
(
γ
+
δ
1
−
1
)
,
(
δ
+
α
1
−
1
)
. Then the value of
a
2
a
0
\frac{a_{2}}{a_{0}}
a
0
a
2
is :
\newline
(a)
4
4
4
\newline
(b)
−
4
-4
−
4
\newline
(c)
6
6
6
\newline
(d) none of these
Get tutor help
Given
P
(
x
)
=
3
x
4
−
2
x
3
−
x
2
−
12
x
−
4
P(x)=3x^{4}-2x^{3}-x^{2}-12x-4
P
(
x
)
=
3
x
4
−
2
x
3
−
x
2
−
12
x
−
4
, write
P
(
x
)
P(x)
P
(
x
)
as a product of linear factors. (
3
3
3
.
4
4
4
/
3
3
3
.
5
5
5
)
\newline
P
(
x
)
=
(
3
x
+
1
)
(
x
−
2
)
(
x
−
(
−
1
2
+
7
2
i
)
)
(
x
−
(
−
1
2
−
7
2
i
)
)
P(x) = \left(3x + 1\right)\left(x - 2\right)\left(x - \left(-\frac{1}{2} + \frac{\sqrt{7}}{2}i\right)\right)\left(x - \left(-\frac{1}{2} - \frac{\sqrt{7}}{2}i\right)\right)
P
(
x
)
=
(
3
x
+
1
)
(
x
−
2
)
(
x
−
(
−
2
1
+
2
7
i
)
)
(
x
−
(
−
2
1
−
2
7
i
)
)
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q
=
9
r
2
+
16
s
2
r
2
q=9 r^{2}+16 s^{2} r^{2}
q
=
9
r
2
+
16
s
2
r
2
\newline
Which of the following equations correctly expresses
r
r
r
in terms of
q
q
q
and
s
s
s
?
\newline
Choose
1
1
1
answer:
\newline
(A)
r
=
±
q
3
+
4
s
r= \pm \frac{\sqrt{q}}{3+4 s}
r
=
±
3
+
4
s
q
\newline
(B)
r
=
±
q
9
+
16
s
2
r= \pm \sqrt{\frac{q}{9+16 s^{2}}}
r
=
±
9
+
16
s
2
q
\newline
(C)
r
=
±
q
25
s
2
r= \pm \sqrt{\frac{q}{25 s^{2}}}
r
=
±
25
s
2
q
\newline
(D)
r
=
±
q
−
16
s
2
3
r= \pm \frac{\sqrt{q-16 s^{2}}}{3}
r
=
±
3
q
−
16
s
2
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