Equations That Lead to a Quadratic

Some equations are not quadratic but can be turned into one. In x⁴ − 5x² + 4 = 0 we put t = x² and get t² − 5t + 4 = 0. In an equation with x in the denominator we multiply through by the denominator. Either way, always check that the roots make sense.

What does this generator do?

On the easy level you solve equations x⁴ + bx² + c = 0 — with four, two or no solutions. On the medium level you solve equations with x in the denominator, such as x − 12/x = 1, and sometimes a root breaks the condition. On the hard level you substitute a whole expression, for example (x² − 3x)² − 2(x² − 3x) − 8 = 0, and solve equations with two fractions.

How to use it

1) For x⁴ + bx² + c = 0, put t = x² and solve the quadratic in t. 2) Go back to x: a positive t gives two solutions ±√t, a negative t gives none. 3) With x in a denominator, first write the conditions (denominator ≠ 0). 4) Multiply through by the denominators and solve. 5) Reject any root that breaks a condition.