Choosing a quadratic substitution An equation may be quadratic in an expression even when it is not quadratic in x. Set u equal to the lower repeated power or expression. The pattern A(x)2+bA(x)+c=0 becomes u2+bu+c=0. The substitution records what u means; solving for u is only an intermediate stage. Identify the repeated expression; state its possible values; solve the quadratic; solve every valid equation A(x)=u; remove duplicates and check the original equation. Example For x6-7x3-8=0, use u=x3, which can be any real number. (u-8)(u+1)=0, so x=2 or x=-1. Check: with u=x2, negative u is impossible over the reals. With u=x3, negative u is allowed. Each positive x2 value gives two signs.

An equation may be quadratic in an expression even when it is not quadratic in \(x\). Set \(u\) equal to the lower repeated power or expression.

The pattern \(A(x)^2+bA(x)+c=0\) becomes \(u^2+bu+c=0\). The substitution records what \(u\) means; solving for \(u\) is only an intermediate stage.

Identify the repeated expression; state its possible values; solve the quadratic; solve every valid equation \(A(x)=u\); remove duplicates and check the original equation.

For \(x^6-7x^3-8=0\), use \(u=x^3\), which can be any real number. \((u-8)(u+1)=0\), so \(x=2\) or \(x=-1\).

Check: with \(u=x^2\), negative \(u\) is impossible over the reals. With \(u=x^3\), negative \(u\) is allowed. Each positive \(x^2\) value gives two signs.