Answer: the correct answer is C:
�
=
1
2
x=
2
1
and
�
=
−
1
8
x=−
8
1
.
Explanation:
To find the zeros of the function
�
(
�
)
=
16
�
2
−
6
�
−
1
f(x)=16x
2
−6x−1, you need to solve for
�
x when
�
(
�
)
=
0
f(x)=0. This means you need to find the values of
�
x that make the equation
16
�
2
−
6
�
−
1
=
0
16x
2
−6x−1=0 true.
You can use the quadratic formula to find the zeros:
�
=
−
�
±
�
2
−
4
�
�
2
�
x=
2a
−b±
b
2
−4ac
In your case,
�
=
16
a=16,
�
=
−
6
b=−6, and
�
=
−
1
c=−1. Plug these values into the quadratic formula:
�
=
−
(
−
6
)
±
(
−
6
)
2
−
4
⋅
16
⋅
(
−
1
)
2
⋅
16
�
=
6
±
36
+
64
32
�
=
6
±
100
32
�
=
6
±
10
32
x=
2⋅16
−(−6)±
(−6)
2
−4⋅16⋅(−1)
x=
32
6±
36+64
x=
32
6±
100
x=
32
6±10
Now, you have two possible solutions:
�
=
6
+
10
32
=
16
32
=
1
2
x=
32
6+10
=
32
16
=
2
1
�
=
6
−
10
32
=
−
4
32
=
−
1
8
x=
32
6−10
=
32
−4
=−
8
1
So, the zeros of the function are
�
=
1
2
x=
2
1
and x = -\frac{1}{8. Therefore, the correct answer is C:
�
=
1
2
x=
2
1
and
�
=
−
1
8
x=−
8
1
.