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

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