If α and β are the roots of the indicial equation 2^(2x+3)=9(2^x )-1,find α+β

Answer 1

2^2x+3 = 9(2^x) - 1

2^2x2³ = 9(2^x) - 1

8(2^2x) = 9(2^x) - 1

8(2^x)² - 9(2^x) + 1 = 0

Let u = 2^x. Then 8u² - 9u + 1 = 0

(8u - 1)(u - 1) = 0

u = 1/8 or u = 1

2^x = 1/8 or 2^x = 1

2^x = 2^-3 or 2^x = 2⁰

So, x = -3 or 0