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
By Quadratic formula
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
By Quadratic formula
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
}

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