Substitution

Biquadratic equations are special 4th degree polynomial equations of the form:

Biquadratic equations can be converted into quadratic equations by substitution $x^2=z$.

Example

Solve the biquadratic equation: $x^4-3x^2+2=0$

1. Substitution

   The given equation is substituted by replacing $x^2$ with $z$.
   $x^4-3x^2+2=0$
   
   $z=x^2$
   $z^2-3z+2=0$

2. Solve the quadratic equation

   The new quadratic equation can now be solved e.g. with the pq-formula.
   $z^2-3z+2=0$
   
   $z_{1,2} = \frac{p}{2} \pm\sqrt{(\frac{p}{2})^2-q}$
   
   $z_{1,2} = \frac32 \pm\sqrt{(\frac32)^2-2}$
   $z_{1,2} = \frac32 \pm\sqrt{\frac14}$
   $z_{1,2} = \frac32 \pm\frac12$
   
   $z_1=2$ and $z_2=1$

3. Backward substitution

   Now you can calculate $x$ from the solutions for $z$.
   To do this, we take the original equation and transform it:
   $x^2=z\quad|\pm\sqrt{}$
   $x=\pm\sqrt{z}$
   
   Use both z-values.
   $x_{1,2}=\pm\sqrt{z_1}$
   $x_1=\sqrt{2}\approx1.41$
   $x_2=-\sqrt{2}\approx-1.41$
   
   $x_{3,4}=\pm\sqrt{z_2}$
   $x_3=\sqrt{1}=1$
   $x_4=-\sqrt{1}=-1$