## Solving Quintic Equations

There are general solutions for quintic polynomials – polynomials of order 4. They may be real or not real depending on the polynomial. We are interested here in a special class of quintic polynomials which factorises into two quadratics which we can solve.

For example, solve Substitute so that and the equation becomes This factorises to give so or 4, hence or 4 so or Example: Solve Substitute to get This factorises to give hence or Using the substitution we have or hence which is impossible or The only solutions are Sometimes you have to be sure that you are square rooting a positive number.

Example This expression does not factorise but we can use the normal quadratic formula to solve for then if the solutions for are positive, we can square root to obtain In the equation   Calculation of these two decimals confirms they are both positive. Hence we can square root them and or Example In the equation   Calculation of these two decimals confirms they are both negative. Hence we cannot square root them there are no real roots for this equation. 