To find the x-intercepts of the quadratic function
f
(
x
)
=
2
x
2
−
4
x
−
4
f(x)=2x
2
−4x−4, you need to solve for
x
x when
f
(
x
)
=
0
f(x)=0. These are the values of
x
x where the graph of the function intersects or crosses the x-axis.
First, set
f
(
x
)
=
0
f(x)=0:
2
x
2
−
4
x
−
4
=
0
2x
2
−4x−4=0
Now, you can solve this quadratic equation. You can either use the quadratic formula or factor the equation. In this case, I'll use the quadratic formula:
x
=
−
b
±
b
2
−
4
a
c
2
a
x=
2a
−b±
b
2
−4ac
In your equation:
a
=
2
a=2,
b
=
−
4
b=−4,
c
=
−
4
c=−4
Substitute these values into the formula:
x
=
−
(
−
4
)
±
(
−
4
)
2
−
4
(
2
)
(
−
4
)
2
(
2
)
x=
2(2)
−(−4)±
(−4)
2
−4(2)(−4)
Simplify:
x
=
4
±
16
+
32
4
x=
4
4±
16+32
x
=
4
±
48
4
x=
4
4±
48
Now, calculate the two possible values for
x
x:
x
1
=
4
+
48
4
x
1
=
4
4+
48
x
2
=
4
−
48
4
x
2
=
4
4−
48
You can further simplify these values:
x
1
=
4
+
4
3
4
=
1
+
3
x
1
=
4
4+4
3
=1+
3
x
2
=
4
−
4
3
4
=
1
−
3
x
2
=
4
4−4
3
=1−
3
So, the x-intercepts (smaller x-value and larger x-value) are
x
1
=
1
+
3
x
1
=1+
3
and
x
2
=
1
−
3
x
2
=1−
3
.