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This essential trains for: AMC-10, AMC-12, AIME, Math Kangaroo 9-10.

Bi-quadratic Equations are the most common:

equation

and the simple change of variable:

equation

equation

turns them into quadratics.

Quartic reciprocal equations have the form:

equation

The approach is to divide by x2:

equation

and regroup to emphasize a well-known pattern:

equation

Use a known algebraic identity:

equation

equation

and substitute:

equation

equation

Make the change of variable:

equation

to obtain the quadratic equation:

equation

Solve and back substitute to find the four solutions.

Translating equations: is often required in order to reduce the size of the coefficients and simplify the aspect of the equation.

Example:

equation

equation

equation

equation

Change of variable to simplify as in the equation:

equation

where the change of variable:

equation

helps simplify the equation to:

equation

which is quadratic.