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This essential trains for: AMC-10, AMC-12, AIME, Math Kangaroo 9-10.
Bi-quadratic Equations are the most common:
and the simple change of variable:
turns them into quadratics.
Quartic reciprocal equations have the form:
The approach is to divide by x2:
and regroup to emphasize a well-known pattern:
Use a known algebraic identity:
Make the change of variable:
to obtain the quadratic 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.
Change of variable to simplify as in the equation:
where the change of variable:
helps simplify the equation to:
which is quadratic.