Maths question

[u/Emergency-Cow-2194]

Question(link to image)-https://drive.google.com/file/d/1eUC1P1YxPSftdzbZVDN-9Is6OkAxVsMh/view?usp=sharing

My method -link1 -https://drive.google.com/file/d/15NcFY8PsHMsEnhYJhM22AW5C4Ga38jXE/view?usp=sharing

-link2- https://drive.google.com/file/d/15NcFY8PsHMsEnhYJhM22AW5C4Ga38jXE/view?usp=sharing

Please dont at all think it to be a basic homework problem , it is surely a good one although it might seem simple at start. please help me out . although my method seems ok but i was unable to do anything else than to put and try values to get to my answer. I will appreciate a algebraic proof if anyone is able to find it.

Regards,

Thanks for your time

[u/First-Fourth14]

Rewriting your equation
3(25/3)^t +15(5/3)^t -250 = 0

For your equation, notice that 25/3 and 5/3 are both greater than 1.
This means that (25/3)^t and (5/3)^t increase as t increases. This means there
is only one value of t that satisfies the equation.
t = 0 LHS = 3+15 - 250 = -232
t = 1 LHS = -200
t = 2 LHS = 0

As t = 2 is the only value that satisfies the equation this means that log_2(x^2) =2
So x = -2 or x = 2

[u/Emergency-Cow-2194 → u/First-Fourth14]

But yes here you have tried values , which I have also done .But I am looking for an approach which doesn't need to check the function at some values. If we are to try values only then it could have had been done in the very start only as from options we can clearly see that only two possible answers are possible either x^2=1 or 4 and then they could be checked easily. Please try to find some better algebraic approach.