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