# Equations Which Are Quadratic in Form

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 .

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