First, I thnk it's clear that you don't mean a quadratic equation, but exactly what you wrote, [imath]3^x+2^x=5[/imath], which is an exponential equation. You probably know how to solve a quadratic equation.

Second, you're asking two different questions, perhaps without realizing it.

If you're expecting **algebraic manipulations** to solve it (that is, to find the entire solution set), as in solving a linear or quadratic equation, that won't work. (Many students don't realize that most equations you can write can't be solved by the methods we teach in school!)

For this sort of equation, the best you can do to **find **the solution is either to graph the equation and see it, or to just recognize that 3+2=5 and know it in a flash of insight (which mathematicians call "solving by inspection"). This part, you've already done.

Then you can **prove that it is** a solution by putting x = 1 into the equation and showing that it is true. This, too, you have already done.

But that only finds **a** solution; you still need to **prove** that it is the **complete **solution. That is where monotonicity (that is, the fact that the function is always increasing, and therefore is one-to-one, comes in.

Now you need to tell us how much you know of algebra and calculus; the latter is the easiest (and maybe only) way to prove this.