# Quintic Equation Solver

Polynomial equations of the form

x^{5} + bx^{4} + cx^{3} + dx^{2} + ex + f = 0

are called quintic equations.

If a quintic equation has real number coefficients, then its five roots follow one of these patterns: five real, three real and two complex, or one real and four complex. Notice that every quintic equation has at least one real root.

Unlike quartics, cubics, and quadratics, for which there exist tidy formulas, the quintic equation does not have any general *quintic formula* that allows you to express the solutions in terms of radicals.

For practical applications, you can use a numerical algorithm to find one of the real roots of the quintic, then use that value to reduce the equation to a quartic. This is the algorithm programmed into the quintic equation solving calculator on the left.

**Example:** Solve the quintic equation x^{5} - 9x^{4} + 25x^{3} - 29x^{2} + 24x - 20 = 0. The first step is to use a numerical algorithm, such as the Newton-Raphson Method, to find a real root of the equation. For instance, if we use the Newton-Raphson Method with a starting guess of 10, we find one solution is x = 5.

Next, we factor (x - 5) out of the quintic to obtain a quartic equation. Doing so yields

x^{4} - 4x^{3} + 5x^{2} - 4x + 4 = 0.

You can now use the quartic formula to solve this equation, whose roots are 2, 2, i, and -i. Therefore, the complete set of quintic roots is {5, 2, 2, i, -i}.

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