Question

The values of x satisfying the equation 5^2x - 5^x+3 + 125 = 5^x are :

(a) 0 and 2 

(b) –1 and 3 

(c) 0 and –3 

(d) 0 and 3

Answer 1

(d) 0 and 3

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

⇒ 5^2x - 5^x x 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.