 Fsc Part 1 Mathematics (Complete Solution)

# Q10 Solve the Following equations (x + 1)(2x + 3)(2x + 5)(x + 3) = 945

(x + 1)(2x + 3)(2x + 5)(x + 3) = 945

Solutions

The given equation can be written as
(x + 1)2(x +
 3 2
)2(x +
 5 2
)(x + 3) - 945 = 0
As
1 +
 3 2
+
 5 2
+ 3
2
=
 16 2
2
= 4, So we can write as

[(x + 1)(x + 3)] [2(x +
 3 2
)2(x +
 5 2
)] - 945 = 0
(x2 + 3x + x + 3) 4(x2 +
 5 2
x +
 3 2
x +
 15 4
) - 945 = 0
(x2 + 4x + 3) 4(x2 + 4x +
 15 4
) - 945 = 0

Let x2 + 4x = t       and Put in the equation

(t + 3) 4(t +
 15 4
) - 945 = 0
 (t + 3) (4t + 15) - 945 = 0
4t2 + 15t + 12t + 45 - 945 = 0
4t2 + 27t - 900 = 0

t =
-27 ±
 √ (27)2 - 4(4)(-900)
2(4)
t =
-27 ±
 √ 729 + 14400
8
t =
-27 ±
 √ 15129
8
t =
-27 ±
 √ (123)2
8
t =
 -27 ± 123
8

Either
t =
 -27 - 123
8
OR
t =
 -27 + 123
8
t =
 -150 8
t =
 96 8
t =
 -75 4
 t = 12

By Letting t = x2 + 4x,       Putting the value of t, we get
x2 + 4x =
 -75 4
 x2 + 4x = 12
4x2 + 16x = -75 x2 + 4x = 12
4x2 + 16x + 75 = 0 --------------------- (i) x2 + 4x - 12 = 0 --------------------- (ii)

By Equation (i)       4x2 + 16x + 75 = 0

Using the Quadratic formula, we have

x =
-16 ±
 √ (16)2 - 4(4)(75)
2(4)
x =
-16 ±
 √ 256 - 1200
8
x =
-16 ±
 √ -944
8
x =
-16 ±
 √ (16)(-59)
8
x =
-16 ± 4
 √ -59
8
x =
4(-4 ±
 √ -59
)
8
x =
-4 ±
 √ -59
2

By Equation (ii)       x2 + 4x - 12 = 0

Using the Quadratic formula, we have

x =
-4 ±
 √ (4)2 - 4(1)(-12)
2(1)
x =
-4 ±
 √ 16 + 48
2
x =
-4 ±
 √ 64
2
x =
 -4 ± 8
2

Either
x =
 -4 - 8
2
OR
x =
 -4 + 8
2
x =
 -12 2
x =
 4 2
 x = -6
 x = 2

Hence the Solution Set of given equation is
{ -6, 2 ,
-4 -
 √ -59
2
,
-4 +
 √ -59
2
}