Question

Find all real numbers satisfying 4^x - 3(2^{x + 2}) + 2⁵ = 0

Answer 1

4^x - 3(2^{x + 2}) + 2⁵ = 0

⇒ (2²)^x – 3(2^x. 2²) + 2⁵ = 0 

(2²)^x – 12.2^x + 32 = 0 

Let y = 2^x 

y² – 12y + 32 = 0 

⇒ (y – 4) (y – 8) = 0 

y – 4 = 0 or y – 8 = 0 

Case (i): 2^x = 4 

⇒ 2^x = (2)² 

⇒ x = 2 

Case (ii): 2x = 8 

⇒ 2^x = (2)³ 

⇒ x = 3 

∴ The roots are 2, 3