Mathematics
Solve :
(x2 + 5x + 4)(x2 + 5x + 6) = 120
Quadratic Equations
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Answer
Let x2 + 5x = a we get,
⇒ (a + 4)(a + 6) = 120
⇒ a2 + 6a + 4a + 24 = 120
⇒ a2 + 10a + 24 - 120 = 0
⇒ a2 + 10a - 96 = 0
⇒ a2 + 16a - 6a - 96 = 0
⇒ a(a + 16) - 6(a + 16) = 0
⇒ (a - 6)(a + 16) = 0
⇒ (a - 6) = 0 or (a + 16) = 0
⇒ a = 6 or a = -16.
Considering a = 6 we get,
⇒ x2 + 5x = 6
⇒ x2 + 5x - 6 = 0
⇒ x2 + 6x - x - 6 = 0
⇒ x(x + 6) - 1(x + 6) = 0
⇒ (x - 1)(x + 6) = 0
⇒ (x - 1) = 0 or (x + 6) = 0
⇒ x = 1 or x = -6.
Considering a = -16 we get,
⇒ x2 + 5x = -16
⇒ x2 + 5x + 16 = 0
In this case,
D = b2 - 4ac = 52 - 4(1)(16) = 25 - 64 = -39 < 0.
It means roots are imaginary in this case.
Hence, x = 1, -6.
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