 Fsc Part 1 Mathematics (Complete Solution)

# Q3 Solve the Following equations

x6 - 9x3 + 8 = 0
Solution

x6 - 9x3 + 8 = 0
Let x3 = t and Put in the given equation
t2 - 9t + 8 = 0
Comparing the given equation with at2 + bt + c = 0, we get
a = 1,
b = -9,
c = 8

By Quardratic formula, we have
t =
-b ±
 √ b2 - 4ac
2a
Putting the values of a, b, and c in the formula

t =
-(-9) ±
 √ (-9)2 - 4(1)(8)
2(1)
t =
9 ±
 √ 81 - 32
2
t =
9 ±
 √ 49
2
t =
9 ±
 √ 72
2
t =
 9 ± 7
2

Either
t =
 9 - 7
2
OR
t =
 9 + 7
2
t =
 2 2
t =
 16 2
 t = 1
 t = 8

By letting x3 = t, Putting the value of t, we get
 x3 = 1 ------------------ (i)
 x3 = 8 ------------------ (ii)

By equation (i)      x3 = 1
x3 - 1 = 0
x3 - 13 = 0
(x - 1)(x2 + x + 1) = 0
Either         x - 1 = 0         OR         x2 + x + 1 = 0
x = 1
x =
-1 ±
 √ (1)2 - 4(1)(1)
2(1)
x =
-1 ±
 √ -3
2

By equation (ii)      x3 = 8
x3 - 8 = 0
x3 - (2)3 = 0
(x - 2)(x2 + 2x + 4) = 0
Either         x - 2 = 0         OR         x2 + 2x + 4 = 0
x = 2
x =
-2 ±
 √ (2)2 - 4(1)(4)
2(1)
x =
-2 ±
 √ 4 - 16
2
x =
-2 ±
 √ -12
2
x =
-2 ±
 √ -3(4)
2
x =
-2 ± 2
 √ -3
2
x =
-1 ±
 √ -3

From equation (i) and (ii), we get
x = 1,
x =
-1 ±
 √ -3
2
, x = 2, and
x =
-1 ±
 √ -3

Hence the Solution Set of given equation is
{ 1 ,     2 ,
-1 -
 √ -3
,
-1 +
 √ -3
,
-1 -
 √ -3
,
-1 +
 √ -3
}