Question

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

Answer 1

Answer:

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