1.

 Solve : (x + 1) (x + 2) (x + 3) (x + 4) + 1 = 0

Answer»

[(x + 1) (x + 4)] [(x + 2) (x + 3)] + 1 = 0            (∵ 1 + 4 = 2 + 3 = 5) 

⇒ (x2 + 5x + 4) (x2 + 5x + 6) + 1 = 0 

Let x2 + 5x = y. Then, (y + 4) (y + 6) + 1 = 0 

⇒ y2 + 10y + 24 + 1 = 0 ⇒ y2 + 10y + 25 = 0 ⇒ (y + 5)2 = 0 ⇒ y = – 5

∴ x2 + 5x = – 5 ⇒ x2 + 5x + 5 = 0. ⇒ x = \(\frac{-5±\sqrt{25-20}}{2}\) = \(\frac{-5±\sqrt{2}}{2}\).


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