Quadratic equations are easy to solve. You can factorise, or failing that, use the quadratic formula. If the quadratic formula returns no real solutions, the quadratic formula has no real solutions.

Many equations can be transformed into quadratic equations by substitution and rearrangement. becomes by substituting  becomes by substituting  becomes on multiplying by and then on substituting The quadratic equation can then be solved in the normal way. can be found by substituting the solution to the quadratic into the substitution made, and solving this to find You may find there are no solutions, one solution or two solutions for the original equation, just as there may be no solutions, one solution or two solutions for the related quadratic. However, just because the quadratic equation has solutions, it does not follow that the original equation has solutions. If the quadratic equation has no solutions however, neither has the original equation.

Example: Solve Substitute to give This expression factorises to give so or To find we use the original substitution solving the two equations and  or Example: Solve Substitute to give This expression factorises to give so or To find we use the original substitution solving the two equations and  or The first solution above does not exist since does not exist.

Example: Solve Substitute to give This expression factorises to give so or To find we use the original substitution solving the two equations and  or The equation has no solutions since neither or exist. 