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Math Problems
Algebra 1
Transformations of absolute value functions: translations and reflections
Ron was asked to determine whether
f
(
x
)
=
1
∣
x
∣
f(x)=\frac{1}{|x|}
f
(
x
)
=
∣
x
∣
1
is even, odd, or neither. Here is his work:
\newline
Step
1
1
1
: Find expression for
f
(
−
x
)
f(-x)
f
(
−
x
)
\newline
f
(
−
x
)
=
1
∣
(
−
x
)
∣
=
1
∣
x
∣
\begin{aligned} f(-x) & =\frac{1}{|(-x)|} \\ & =\frac{1}{|x|} \end{aligned}
f
(
−
x
)
=
∣
(
−
x
)
∣
1
=
∣
x
∣
1
\newline
Step
2
2
2
: Check if
f
(
−
x
)
f(-x)
f
(
−
x
)
is equal to
f
(
x
)
f(x)
f
(
x
)
or
−
f
(
x
)
-f(x)
−
f
(
x
)
\newline
1
∣
x
∣
\frac{1}{|x|}
∣
x
∣
1
is the same as
f
(
x
)
=
1
∣
x
∣
f(x)=\frac{1}{|x|}
f
(
x
)
=
∣
x
∣
1
.
\newline
Step
3
3
3
: Conclusion
\newline
f
(
−
x
)
f(-x)
f
(
−
x
)
is equivalent to
f
(
x
)
f(x)
f
(
x
)
, so
f
f
f
is odd.
\newline
Is Ron's work correct? If not, what is the first step where Ron made a mistake?
\newline
Choose
1
1
1
answer:
\newline
(A) Ron's work is correct.
\newline
(B) Ron's work is incorrect. He first made a mistake in Step
1
1
1
.
\newline
(C) Ron's work is incorrect. He first made a mistake in Step
2
2
2
.
\newline
(D) Ron's work is incorrect. He first made a mistake in Step
3
3
3
.
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Mice
A
A
A
and
B
B
B
each went on a quest to find food. Their displacements by the end are represented by vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, respectively.
\newline
Which option best describes the meaning of the following statement?
\newline
∥
a
⃗
∥
=
∥
b
⃗
∥
\| \vec{a} \| = \| \vec{b} \|
∥
a
∥
=
∥
b
∥
\newline
Choose
1
1
1
answer:
\newline
(A) The mice finished at the same distance from their starting points, but not necessarily in the same direction.
\newline
(B) The mice finished at the same distance and in the same direction from their starting points.
\newline
(C) The mice walked the exact same path.
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
8
,
8
)
(8,8)
(
8
,
8
)
and a terminal point at
(
8
,
6
)
(8,6)
(
8
,
6
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
1
,
4
)
(-1,4)
(
−
1
,
4
)
and a terminal point at
(
1
,
6
)
(1,6)
(
1
,
6
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
1
,
4
)
(-1,4)
(
−
1
,
4
)
and a terminal point at
(
0
,
1
)
(0,1)
(
0
,
1
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
0
,
6
)
(0,6)
(
0
,
6
)
and a terminal point at
(
2
,
0
)
(2,0)
(
2
,
0
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
1
,
3
)
(-1,3)
(
−
1
,
3
)
and a terminal point at
(
−
2
,
4
)
(-2,4)
(
−
2
,
4
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
7
,
6
)
(7,6)
(
7
,
6
)
and a terminal point at
(
1
,
0
)
(1,0)
(
1
,
0
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
3
,
0
)
(-3,0)
(
−
3
,
0
)
and a terminal point at
(
−
5
,
−
5
)
(-5,-5)
(
−
5
,
−
5
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
2
,
−
6
)
(-2,-6)
(
−
2
,
−
6
)
and a terminal point at
(
−
1
,
−
4
)
(-1,-4)
(
−
1
,
−
4
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
7
,
4
)
(-7,4)
(
−
7
,
4
)
and a terminal point at
(
−
8
,
4
)
(-8,4)
(
−
8
,
4
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
2
,
−
3
)
(-2,-3)
(
−
2
,
−
3
)
and a terminal point at
(
−
5
,
−
2
)
(-5,-2)
(
−
5
,
−
2
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
7
,
2
)
(7,2)
(
7
,
2
)
and a terminal point at
(
2
,
6
)
(2,6)
(
2
,
6
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
0
,
2
)
(0,2)
(
0
,
2
)
and a terminal point at
(
−
1
,
2
)
(-1,2)
(
−
1
,
2
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
8
,
8
)
(8,8)
(
8
,
8
)
and a terminal point at
(
8
,
2
)
(8,2)
(
8
,
2
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
3
,
4
)
(-3,4)
(
−
3
,
4
)
and a terminal point at
(
−
6
,
4
)
(-6,4)
(
−
6
,
4
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
1
,
0
)
(-1,0)
(
−
1
,
0
)
and a terminal point at
(
2
,
−
3
)
(2,-3)
(
2
,
−
3
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
3
,
−
4
)
(3,-4)
(
3
,
−
4
)
and a terminal point at
(
6
,
−
1
)
(6,-1)
(
6
,
−
1
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
1
,
−
8
)
(-1,-8)
(
−
1
,
−
8
)
and a terminal point at
(
0
,
−
5
)
(0,-5)
(
0
,
−
5
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
2
,
0
)
(-2,0)
(
−
2
,
0
)
and a terminal point at
(
−
4
,
−
3
)
(-4,-3)
(
−
4
,
−
3
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
0
,
−
3
)
(0,-3)
(
0
,
−
3
)
and a terminal point at
(
1
,
1
)
(1,1)
(
1
,
1
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
4
,
4
)
(4,4)
(
4
,
4
)
and a terminal point at
(
1
,
7
)
(1,7)
(
1
,
7
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
5
,
1
)
(5,1)
(
5
,
1
)
and a terminal point at
(
4
,
−
4
)
(4,-4)
(
4
,
−
4
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
1
,
0
)
(1,0)
(
1
,
0
)
and a terminal point at
(
1
,
−
6
)
(1,-6)
(
1
,
−
6
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
−
5
,
6
)
(-5,6)
(
−
5
,
6
)
and a terminal point at
(
−
6
,
1
)
(-6,1)
(
−
6
,
1
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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(
x
−
4
)
(
x
−
5
)
=
0
(x-4)(x-5)=0
(
x
−
4
)
(
x
−
5
)
=
0
\newline
If
x
=
s
x=s
x
=
s
and
x
=
t
x=t
x
=
t
are the solutions to the given equation, which of the following is equal to the value of
∣
s
−
t
∣
|s-t|
∣
s
−
t
∣
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
9
-9
−
9
\newline
(B)
−
1
-1
−
1
\newline
(C)
1
1
1
\newline
(D)
9
9
9
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Eva sees a dolphin
3
3
3
.
2
2
2
meters below sea level and a bird
47
10
\frac{47}{10}
10
47
meters above sea level.
\newline
Which
2
2
2
of the following expressions represents the vertical distance between the dolphin and the bird?
\newline
Choose
2
2
2
answers:
\newline
(A)
∣
47
10
+
3.2
∣
\left|\frac{47}{10}+3.2\right|
∣
∣
10
47
+
3.2
∣
∣
\newline
(B)
∣
−
3.2
−
47
10
∣
\left|-3.2-\frac{47}{10}\right|
∣
∣
−
3.2
−
10
47
∣
∣
\newline
(C)
∣
3.2
−
47
10
∣
\left|3.2-\frac{47}{10}\right|
∣
∣
3.2
−
10
47
∣
∣
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We want to solve the following equation.
\newline
∣
x
−
4
∣
=
x
2
−
6
x
+
9
|x-4|=x^{2}-6 x+9
∣
x
−
4∣
=
x
2
−
6
x
+
9
\newline
One of the solutions is
x
≈
1.4
x \approx 1.4
x
≈
1.4
.
\newline
Find the other solution.
\newline
Hint: Use a graphing calculator.
\newline
Round your answer to the nearest tenth.
\newline
x
≈
x \approx
x
≈
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Fill in the blank.
\newline
0.
7
−
−
∣
−
6.7
∣
0.7_{--}|-6.7|
0.
7
−−
∣
−
6.7∣
\newline
Choose
1
1
1
answer:
\newline
(A)
<
<
<
\newline
(B)
>
>
>
\newline
(c)
=
=
=
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What is the modulus (absolute value) of
15
+
8
i
15+8 i
15
+
8
i
?
\newline
Don't round. If necessary, express your answer as a radical.
\newline
∣
15
+
8
i
∣
=
|15+8 i|=
∣15
+
8
i
∣
=
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What is the modulus (absolute value) of
7
−
i
7-i
7
−
i
?
\newline
Don't round. If necessary, express your answer as a radical.
\newline
∣
7
−
i
∣
=
|7-i|=
∣7
−
i
∣
=
Get tutor help
What is the modulus (absolute value) of
−
12
+
5
i
-12+5 i
−
12
+
5
i
?
\newline
Don't round. If necessary, express your answer as a radical.
\newline
∣
−
12
+
5
i
∣
=
|-12+5 i|=
∣
−
12
+
5
i
∣
=
Get tutor help
What is the modulus (absolute value) of
−
6
+
4
i
-6+4 i
−
6
+
4
i
?
\newline
Don't round. If necessary, express your answer as a radical.
\newline
∣
−
6
+
4
i
∣
=
|-6+4 i|=
∣
−
6
+
4
i
∣
=
Get tutor help
Particles
A
\mathrm{A}
A
and
B
\mathrm{B}
B
are moving along a plane. Their velocities are represented by vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, respectively.
\newline
Which option best describes the meaning of the following statement?
\newline
∥
a
⃗
∥
>
∥
b
⃗
∥
\|\vec{a}\|>\|\vec{b}\|
∥
a
∥
>
∥
b
∥
\newline
Choose
1
1
1
answer:
\newline
(A) Particle
A
A
A
is heavier than particle
B
B
B
.
\newline
(B) Particle A's speed is greater than particle B's speed.
\newline
(C) Particle A's movement creates a greater counterclockwise angle with the eastward direction than particle B's movement.
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Teams
A
\mathrm{A}
A
and
B
\mathrm{B}
B
are playing a game of tug of war. The forces in which they pull at the rope are represented by vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, respectively.
\newline
Which option best describes the meaning of the following statement?
\newline
∥
a
⃗
∥
=
2
∣
∣
b
⃗
∣
∣
\|\vec{a}\|=2|| \vec{b}||
∥
a
∥
=
2∣∣
b
∣∣
\newline
Choose
1
1
1
answer:
\newline
(A) Team A has twice as many people as team
B
B
B
.
\newline
(B) Team A pulls twice as strongly as team B.
\newline
(C) Team A is twice as far from the center of the rope as team B.
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Mice
A
A
A
and
B
B
B
each went on a quest to find food. Their displacements by the end are represented by vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, respectively.
\newline
Which option best describes the meaning of the following statement?
\newline
∥
a
⃗
∥
=
∥
b
⃗
∥
\|\vec{a}\|=\|\vec{b}\|
∥
a
∥
=
∥
b
∥
\newline
Choose
1
1
1
answer:
\newline
(A) The mice finished at the same distance from their starting points, but not necessarily in the same direction.
\newline
(B) The mice finished at the same distance and in the same direction from their starting points.
\newline
(C) The mice walked the exact same path.
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What happens to the value of the expression
q
20
\frac{q}{20}
20
q
as
q
q
q
decreases?
\newline
Choose
1
1
1
answer:
\newline
(A) It increases.
\newline
(B) It decreases.
\newline
(C) It stays the same.
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What happens to the value of the expression
10
d
\frac{10}{d}
d
10
as
d
d
d
increases from a small positive number to a large positive number?
\newline
Choose
1
1
1
answer:
\newline
(A) It increases.
\newline
(B) It decreases.
\newline
(C) It stays the same.
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What happens to the value of the expression
5
x
+
5
\frac{5}{x}+5
x
5
+
5
as
x
x
x
decreases from a large positive number to a small positive number?
\newline
Choose
1
1
1
answer:
\newline
(A) It increases.
\newline
(B) It decreases.
\newline
(C) It stays the same.
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What happens to the value of the expression
c
+
2
c+2
c
+
2
as
c
c
c
increases?
\newline
Choose
1
1
1
answer:
\newline
(A) It increases.
\newline
(B) It decreases.
\newline
(C) It stays the same.
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What happens to the value of the expression
50
p
\frac{50}{p}
p
50
as
p
p
p
decreases from a large positive number to a small positive number?
\newline
Choose
1
1
1
answer:
\newline
(A) It increases.
\newline
(B) It decreases.
\newline
(C) It stays the same.
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What happens to the value of the expression
b
−
1
b-1
b
−
1
as
b
b
b
increases?
\newline
Choose
1
1
1
answer:
\newline
(A) It increases.
\newline
(B) It decreases.
\newline
(C) It stays the same.
Get tutor help
The graph of a sinusoidal function intersects its midline at
(
0
,
5
)
(0,5)
(
0
,
5
)
and then has a maximum point at
(
π
,
6
)
(\pi, 6)
(
π
,
6
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
f(x)=
f
(
x
)
=
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k
x
2
+
5
x
=
−
7
k x^{2}+5 x=-7
k
x
2
+
5
x
=
−
7
\newline
In the given equation,
k
k
k
is a constant. Which of the following represents all values of
k
k
k
for which the equation has two distinct real solutions?
\newline
Choose
1
1
1
answer:
\newline
(A)
k
<
−
25
28
k<-\frac{25}{28}
k
<
−
28
25
\newline
(B)
k
>
−
25
28
k>-\frac{25}{28}
k
>
−
28
25
and
k
≠
0
k \neq 0
k
=
0
\newline
(C)
k
<
25
28
k<\frac{25}{28}
k
<
28
25
and
k
≠
0
k \neq 0
k
=
0
\newline
(D)
k
>
25
28
k>\frac{25}{28}
k
>
28
25
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g
(
w
)
=
(
w
+
13
)
3
(
w
+
19
)
2
g(w)=(w+13)^{3}(w+19)^{2}
g
(
w
)
=
(
w
+
13
)
3
(
w
+
19
)
2
\newline
The polynomial function
g
g
g
is defined. When
g
(
w
)
g(w)
g
(
w
)
is divided by
(
w
+
16
)
(w+16)
(
w
+
16
)
, the remainder is
r
r
r
. What is the value of
∣
r
∣
|r|
∣
r
∣
?
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(
3
y
−
2
)
(
y
+
a
)
=
3
y
2
+
b
z
(3 y-2)(y+a)=3 y^{2}+b z
(
3
y
−
2
)
(
y
+
a
)
=
3
y
2
+
b
z
\newline
If the given equation is true for all values of
y
y
y
, where
a
a
a
and
b
b
b
are constants, which of the following is the value of
b
b
b
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
38
-38
−
38
\newline
(B)
12
12
12
\newline
(C)
34
34
34
\newline
(D)
36
36
36
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Find the zeros of the function. Enter the solutions from least to greatest.
\newline
f
(
x
)
=
(
x
−
4
)
2
−
25
f(x)=(x-4)^{2}-25
f
(
x
)
=
(
x
−
4
)
2
−
25
\newline
lesser
x
=
x=
x
=
\newline
greater
x
=
x=
x
=
Get tutor help
Find the zeros of the function. Enter the solutions from least to greatest.
\newline
f
(
x
)
=
(
x
−
7
)
2
−
64
f(x)=(x-7)^{2}-64
f
(
x
)
=
(
x
−
7
)
2
−
64
\newline
lesser
x
=
x=
x
=
\newline
greater
x
=
x=
x
=
Get tutor help
Find the zeros of the function. Enter the solutions from least to greatest.
\newline
f
(
x
)
=
(
x
−
10
)
2
−
49
f(x)=(x-10)^{2}-49
f
(
x
)
=
(
x
−
10
)
2
−
49
\newline
lesser
x
=
x=
x
=
\newline
greater
x
=
x=
x
=
Get tutor help
x
2
+
k
x
−
14
=
0
x^{2}+k x-14=0
x
2
+
k
x
−
14
=
0
\newline
In the given equation,
k
k
k
is a constant. The equation has solutions at
7
7
7
and
−
2
-2
−
2
. What is the value of
k
k
k
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
9
-9
−
9
\newline
(B)
−
5
-5
−
5
\newline
(C)
5
5
5
\newline
(D)
9
9
9
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2
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