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Byju's Answer
Standard XII
Mathematics
Higher Order Equations
Solve the equ...
Question
Solve the equation
(
x
+
2
)
(
x
+
3
)
(
x
+
8
)
(
x
+
12
)
=
4
x
2
.
.
.
.
.
.
.
.
(
1
)
find the difference between the rational roots .
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Solution
Since
(
−
2
)
(
−
12
)
=
(
−
3
)
(
−
8
)
we can
write equation (1) as
(
x
+
2
)
(
x
+
12
)
(
x
+
3
)
(
x
+
8
)
=
4
x
2
⇒
(
x
2
+
14
x
+
24
)
(
x
2
+
11
x
+
24
)
=
4
x
2
...(2)
Now
x
=
0
is not a root of (1)
Dividing by
x
2
in both sides of (2), we get
⇒
(
x
+
24
x
+
14
)
(
x
+
24
x
+
11
)
=
4
...(3)
Putting
x
+
24
x
=
y
,
we get
(
y
+
14
)
(
y
+
11
)
=
4
⇒
y
2
=
25
y
+
150
=
0
⇒
y
=
−
15
or
y
=
−
10
Thus the ariginal equation is equivlent to
x
+
24
x
=
−
15
or
x
+
24
x
=
−
10
⇒
x
2
+
15
x
+
24
=
0
or
x
2
+
10
x
+
24
=
0
⇒
x
=
−
15
±
√
129
2
or
x
=
−
6
,
−
4
∴
Difference between sational roots is
−
4
−
(
−
6
)
=
2.
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