Slove: 2^2x+3 - 57=65 (2^x-1)

3 and - 3

Step-by-step explanation:

hello : 2^ (2x+3) - 57=65 (2^x-1)

look : 2^2x = (2^x) ² and 2^ (2x+3) = 2^2x * 2^3 = 8 (2^x) ²

now let; 2^x = t ... (t > 0)

8t² - 57 = 65 (t - 1)

8t² - 65t + 8 = 0

quadratic equation : Δ = b²-4ac when : a = 8 and b = - 65 and c = 8

Δ = (-65) ²-4 (8) (8) = 4225-256=3969 = 63²

√Δ = 63

t1 = (-b-√Δ) / 2a = (65-63) / 16 = 1/8

t1 = (-b+√Δ) / 2a = (65+63) / 16 = 128/16 = 16/2 = 8

look : 8 = 2^3 and 1/8 = (2^-1) ^3 and 1/2 = 2^-1

but : 2^x = t ... (t > 0)

2^x = 2^3 ... x=3

2^x = (2^-1) ^3 = 2^-3 ... x = - 3