Non-polynomial Equations

Some equations aren't polynomial equations at all — yet a well-chosen substitution converts them into one that genuinely is, which can then be solved by every tool above.

Worked illustration (radical equation). $\sqrt{2x+3}=x$ is not a polynomial equation (the left side has a square root of an $x$-dependent quantity). Squaring both sides converts it into the genuine polynomial equation $2x+3=x^2$, i.e. $x^2-2x-3=0=(x-3)(x+1)$, giving candidates $x=3$ and $x=-1$. Checking back in the original equation: $x=3$ gives $\sqrt9=3$ ✓ (genuine solution), but $x=-1$ gives $\sqrt1=1\ne-1$ ✗ — squaring silently flipped the sign requirement, manufacturing an extraneous root that solves the squared (polynomial) equation but not the original one. Every candidate produced this way must be checked back in the original equation before being accepted.

Worked illustration (Example 3.29 — trigonometric, polynomial in $\cos x$). Solve $2\cos^2x-9\cos x+4=0$. Substituting $y=\cos x$ gives the genuine quadratic $2y^2-9y+4=0$, with roots $y=4$ and $y=\tfrac12$. Now translate back: $\cos x=4$ is never possible (since $-1\le\cos x\le1$) and must be discarded; $\cos x=\tfrac12$ has infinitely many solutions, $x=2n\pi\pm\tfrac\pi3$ for every integer $n$.