Answers and Explanations 141
Answers
1–100
42.
x = − ±
2
10
3
The equation 3x
2
+ 4x – 2 = 0 doesn’t factor nicely, so use the quadratic equation to
solve for x. Here, a = 3, b = 4, and c = –2:
x
b
b
ac
a
=
− ±
−
=
− ±
− ( ) −
( )
( )
=
− ±
+
=
− ±
= − ±
=
2
2
4
2
4
4 4 3
2
2 3
4
16 24
6
4
40
6
4 2 10
6
− − ±
=
− ±
2
3
10
3
2
10
3
43.
x = −
−
2
5
5
5
,
The equation x
10
+ 7x
5
+ 10 = 0 isn’t quadratic, but by using a substitution, you can
produce a quadratic equation:
x
x
x
x
10
5
5
2
5
7
10 0
7
1 0 0
+
+ =
( ) + ( ) + =
Now use the substitution y = x
5
to form a quadratic equation that you can easily factor:
y
y
y
y
2
7
10 0
2
5 0
+
+ =
+
(
) +
(
)=
From y + 2 = 0, you have the solution y = –2, and from y + 5 = 0, you have the solution
y = –5.
Now replace y using the original substitution and solve for x to get the final answer:
x
5
= –2 gives you x = −2
5
, and x
5
= –5 gives you x = −5
5
.
44.
x = 0, − 5
3
, 1
Begin by factoring out the greatest common factor, x
2
. Then factor the remaining
quadratic expression:
3
2
5
0
3
2
5 0
3
5
1 0
4
3
2
2
2
2
x
x
x
x
x
x
x
x
x
+
−
=
+
−
(
) =
+
− =
(
)(
)
Next, set each factor equal to zero and solve for x: x
2
= 0 has the solution x = 0,
3x + 5 = 0 has the solution x = − 5
3
, and x – 1 = 0 has the solution x = 1.