# 8. Solving difficult equations

The Youtube channel SyberMath has several interesting equations, including rational, irrational, and transcendental. Here we will show some of them so that we can solve them together.

Repeated viewing of technique used should get us to the point where we know how to tackle them.

The channel is at https://www.youtube.com/@SyberMath

## 8.1. Polynomial equations

\[\begin{split}x^6 = (x-1)^6
\\
(a+b)^5 = a^5 + b^5\end{split}\]

## 8.2. Rational equations

Simplify:

\[\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}} + \sqrt{5}\]

## 8.3. Irrational equations

\[\sqrt{x+8} - \sqrt{x} = 2\]

## 8.4. Transcendental equations

\[\begin{split}(\ln x)^5 = \ln x^5
\\
2^{x^{2}} = 3^x
\\
(x^2 - x - 1)^{x+2} = 1\end{split}\]

## 8.5. Functional equations

\[\begin{split}f\left(\frac{2x-1}{x-3}\right) = x^2
\\
x = \frac{3y-1}{y-2}\end{split}\]