## Solving Quadratic Exponential Equations By Substitution

Some exponential equations can be factorised in linear factors. The simplest can be factorised into quadratic equations. We then put each factor equal to zero and solve it.

Example: Solve (1)

Factorise to get  or The above equation has two solutions. In general, as for quadratic equations, an exponential which can be expressed as two factors can have one, two or no solutions. It is convenient to make clear the connection by expressing the original equation as a quadratic using the substitution Then and equation (1) above becomes This equation factorises to give so Since the original equation was expressed in terms of we still have to find but we can use the substitution with the values of that we have found, to find  or Example: Solve Substitute to get and factorise this expression to give  This has no solution since there is no real log of a negative number. Example: Solve Substitute to get and factorise this expression to give  This has no solution since there is no real log of a negative number. This has no solution since there is no real log of a negative number hence the equation has no solutions. 