Equations Reducible to Quadratic Form

An equation "reducible to quadratic form" is simply one that becomes an ordinary quadratic once you rename part of it. The formula up top looks abstract, but it's one move: $x^{2n}$ (the biggest power) is just $(x^n)^2$, so if you call $x^n$ by a new name, $u$, the whole thing turns into $au^2 + bu + c = 0$. You already know three ways to solve that. It's a small trick with a big payoff.

What to look for. You need one power to be exactly twice another, because then the bigger one is the square of the smaller: $x^4$ and $x^2$, $x^{2/3}$ and $x^{1/3}$, $e^{2x}$ and $e^x$. That's what makes the renaming work, since $x^4 = (x^2)^2$ and $e^{2x} = (e^x)^2$.

Going back. Solving for $u$ is only halfway. Each value of $u$ gives an equation for $x$, like $x^2 = 4$, and that can give two answers, one or none. Throw out the impossible ones: $x^2 = -3$ has no real solutions, and $e^x = -1$ has none at all, since $e^x$ is always positive.

A common mistake. It's tempting to stop once you've found $u$ and report $u = 1$ and $u = 4$ as the answers. Those are values of $x^2$, not of $x$. In $x^4 - 5x^2 + 4 = 0$ they turn into four solutions, $\pm1$ and $\pm2$.

Biquadratic equations appear in optics and in the eigenvalues of coupled oscillators; exponential versions appear in finance and population models.