= Quintic equation
A <quintic equation> is a one-variable <polynomial equation>
$$
a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0,\qquad a_5\ne0.
$$
Over the <complex numbers>, it has five <roots of a polynomial> counted with <multiplicity of a root>, by the <fundamental theorem of algebra>. Over the <real numbers>, at least one root is real because the leading odd-degree term makes the <polynomial> take opposite signs far enough in the two directions.
Some quintics are easy to solve: $x^5-x=x(x^2-1)(x^2+1)$ has roots $0,1,-1,i,-i$. The <Abel-Ruffini theorem> says that a formula by radicals cannot solve the general <quintic equation>; it does not rule out special factorizations, other special-function descriptions, or numerical <root finding>. Thus failure of a universal radical formula does not prevent exact or approximate solutions of particular equations.
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