What is the solution? Factoring and computing the equation.

Question

If you will be gracious enough to answer, the equation is currently: $$10^x + 15^{x-1}= 20,$$ What is the value of $2x^2$? Please list all steps, if you don't mind. To follow up, what is the name of the website that lists tags for mathematical signs on this website?

Answer 1

I believe (unfortunately, without evidence) that the given equation is a transcendental equation, and am therefore resolving to employ the numerical Newton-Rhapson method, by finding the root(s) of $10^x + 15^{x-1} - 20 = 0$.

Set $$f(x) = 10^x + 15^{x-1} - 20$$ Then, $$f'(x)=10^x\ln10+15^{x-1}\ln15$$ Define the sequence $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ Then the Newton-Rhapson method says that $x_n$ gives a root of $f(x) = 0$ for some sufficiently large $n$, with an initial guess $x_0$.

We take $x_0=1$ (a random guess), and find that the first few values of $x_n$ are: $$x_1 \approx 1.34932326879814\\ x_2 \approx 1.26510504583041\\ x_3 \approx 1.25547389505072\\ x_4 \approx 1.25535183049781\\ x_5 \approx 1.25535166963961\\ x_6 \approx 1.25535166945069\\ x_7 \approx 1.25535166945047\\ x_8 \approx 1.25535166945047\\ x_9 \approx 1.25535166945047$$ Thus, we see that $10^x + 15^{x-1} - 20 = 0$, or $10^x + 15^{x-1} = 20$ when $x\approx1.25535166945047$.