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

Substituteso thatand the equation becomesThis factorises to givesoor 4, henceor 4 soor

Example: Solve

Substituteto getThis factorises to givehence orUsing the substitutionwe haveorhencewhich is impossible orThe 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 forare 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 andor

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.