1.1.5—Quadratic-form equations

If an equation contains x² and x⁴, set u=x² and solve the resulting quadratic in u. Only values u≥0 correspond to real x, and each positive u gives x=±√u.

Make the substitution explicit, solve in the new variable, apply its domain restriction, then reverse the substitution and check the original equation.

x⁴−5x²+4=0 becomes u²−5u+4=0, so u=1 or 4 and x=±1,±2.

A negative solution for u cannot produce a real x, and replacing x² by u does not mean x itself equals u.