## Quartics as Quadratics

Many equations not actually quadratic equations can be written as quadratics, solved as quadratics and these solution used to solve the original equation. Some quintics - polynomials of degree four are especially suitable for this treatment.We can write the equation

\[x^4-8x^2+12=0 \]

by substituting \[y= x^2\]

. The equation becomes \[y^2-8y+12=0\]

.This equation factorises as

\[(y-6)(y-2)=0\]

.Set each factor equal to 0 and solve.

\[y-6=0 \rightarrow y=6\]

\[y-2=0 \rightarrow y=2\]

\[y=6\]

then \[x^2=6 \rightarrow x= \pm \sqrt{6}\]

.\[y=2\]

then \[x^2 =2 \rightarrow x= \pm \sqrt{2}\]

.