Question

The equation $(x + 1)^{\log_{(x + 1)}(3 + 2x - x^{2})}$= (x – 3) |x| has solutions

• A. 0
• B. 2
• C. 3
• D. 1

Answer 1

Answer: (A)

0

Answer 2

Given equation is

_(x+1)log_{(x +1)} (3 + 2x – x²) = (x – 3) |x| .....(1)

Clearly 3 + 2x – x² > 0 [log x is defined if x > 0]

⇒ x² – 2x – 3 < 0 ⇒ (x – 3) (x + 1) < 0

⇒ – 1 < x < 3 ...(2)

Let x + 1 ∴ x > 0 ....(3)

∴ (1) gives

_(x+1)log_x+1 (3 + 2x – x²) = (x – 3)x

⇒ 3 + 2x – x² = x² – 3x

⇒ 2x² – 5x – 3 = 0 ⇒ x = 3, – $\frac{1}{2}$

But from (2), (3) – 1 < x < 3 and x > 0

∴ x = 3, – $\frac{1}{2}$ are not possible.

Hence there is no solution