Sometimes an equation is not quadratic as is, but becomes quadratic
if you make a * substitution*. Then you can solve
the resulting quadratic, and get solutions to the original equation
by using the substitution equation. Equations of this type are said
to be * quadratic in form*.

* Example.* Solve . (Complex solutions are allowed.)

Write the equation as

The equation is actually quadratic; you can see this more clearly by substituting y for :

Factor and solve:

gives , or .

gives , or .

The solutions are and .

* Example.* Solve . (Complex solutions are
allowed.)

Let . The equation becomes

Factor and solve:

gives , or .

gives , or .

The solutions are or .

* Example.* Solve the equation . (Complex solutions are
allowed.)

Write the equation as

Let . Then

* Example.* Solve the equation . (Complex solutions are
allowed.)

Write the equation as

Let . Then

* Example.* Solve the equation . (Complex solutions are
allowed.)

Let . Then

* Example.* Solve . (Complex solutions are allowed.)

Write the equation as

Let . The equation becomes

Factor and solve:

gives , or .

gives , or .

The solutions are and .

Note: If the problem had asked for only *real* solutions, then
the only solutions are .

Copyright 2016 by Bruce Ikenaga