Sum of the roots of the equation

4^x - 3(2^{x + 3}) + 128 = 0 is

Concept:

Base Rule

If b raised to the xth power is equal to b raised to the yth power, that implies that x = y.” 

\(\rm b^x = b^y \) ⇒ x = y

Calculations:

Given equation is 4x - 3(2x + 3) + 128 = 0

⇒ \(\rm (2^2)^x - 3 (2^x.2^3) + 128 = 0\)

⇒ \(\rm (2^x)^2 - 24 (2^x) + 128 = 0\)

⇒ \(\rm (2^x)^2 - 16 (2^x) - 8(2^x)+ 128 = 0\)

⇒ \(\rm (2^x - 16)(2^x - 8) = 0\)

⇒ \(\rm 2^x = 16 \;\;\text{or}\;\; 2^x = 8\)

⇒ \(\rm 2^x = 2^4 \;\;\text{or}\;\; 2^x = 2^3\)

⇒ x = 4 or x = 3

The roots of the equation 4x - 3(2x + 3) + 128 = 0 are 4 and 3

Its Sum = 4 + 3 = 7