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.

becomesby substituting

becomesby substituting

becomeson multiplying byand thenon 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 findYou 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

Substituteto giveThis expression factorises to givesoor

To findwe use the original substitutionsolving the two equationsand

or

Example: Solve

Substituteto giveThis expression factorises to givesoor

To findwe use the original substitutionsolving the two equationsand

or

The first solution above does not exist sincedoes not exist.

Example: Solve

Substituteto giveThis expression factorises to givesoor

To findwe use the original substitutionsolving the two equationsand

or

The equation has no solutions since neitherorexist.