The values of x satisfying the equation 52x - 5x+3 + 125 = 5x are :(

1.	The values of x satisfying the equation 52x - 5x+3 + 125 = 5x are :(a) 0 and 2 (b) –1 and 3 (c) 0 and –3 (d) 0 and 3
Answer» (d) 0 and 3 5^2x - 5^x+3 + 125 = 5^x ⇒ 5^2x - 5^xx 5³ + 125 = 5^x ⇒ 5^2x - 5^x(125 + 1) + 125 = 0 ⇒ 5^2x - 126 x 5^x + 125 = 0 Let 5^x = y Then, the given equation reduces to y²- 126y + 125 = 0 ⇒ y² - 125y -y + 125 = 0 ⇒ y(y - 125) - 1(y - 125 ) = 0 ⇒ (y - 125) (y - 1) = 0 ⇒ y - 125 = 0 or y - 1 = 0 ⇒ y = 125 or 1 ⇒ 5^x = 125 or 5^x = 1 ⇒ 5^x = 5³ or 5^x = 5⁰ ⇒ x = 3, 0.

The values of x satisfying the equation 52x - 5x+3 + 125 = 5x are :(a) 0 and 2 (b) –1 and 3 (c) 0 and –3 (d) 0 and 3

Answer»

(d) 0 and 3

5^2x - 5^x+3 + 125 = 5^x

⇒ 5^2x - 5^xx 5³ + 125 = 5^x

⇒ 5^2x - 5^x(125 + 1) + 125 = 0

⇒ 5^2x - 126 x 5^x + 125 = 0

Let 5^x = y

Then, the given equation reduces to

y²- 126y + 125 = 0

⇒ y² - 125y -y + 125 = 0

⇒ y(y - 125) - 1(y - 125 ) = 0

⇒ (y - 125) (y - 1) = 0

⇒ y - 125 = 0 or y - 1 = 0

⇒ y = 125 or 1 ⇒ 5^x = 125 or 5^x = 1

⇒ 5^x = 5³ or 5^x = 5⁰ ⇒ x = 3, 0.