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 findwe use the original substitution
solving the two equations
and 
or 
The first solution above does not exist since
does not exist.
Example: Solve
Substituteto give
This expression factorises to give
so
or
To findwe use the original substitutionsolving the two equations
and 
or 
The equation has no solutions since neither
or
exist.