Answer $9^x = 4^x + 6^x$ to a 10th grader, who knows math until the equation of a straight line (just before calculus)

Question

Find $x$: $$9^x = 4^x + 6^x$$

This was in my exam today, and I have no idea, would really help if someone taught me, I would love to know how to write math on this website and is there a way to like search a topic for a 10th grader where i can find many questions and solve them?

Answer 1

$$9^x = 4^x + 6^x$$

Dividing by $4^x$ , we get :

$$\left(\frac{9}{4}\right)^x = 1 + \left(\frac{3}{2}\right)^x$$

$$\left(\frac{3}{2}\right)^{2x} = 1 + \left(\frac{3}{2}\right)^{x}$$

Substituting $\left(\frac{3}{2}\right)^x = t $ , you get a quadratic equation in $t$, $$t^2 = 1 + t$$

Now, solve for $t$ and then solve for $x$.