Math Solve equations Substitution


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


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



The quadratic equation resulting from the substitution can be solved with the quadratic formula.


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

  1. Substitution

    The given equation is substituted by replacing $x^2$ with $z$.

  2. Solve the quadratic equation

    The new quadratic equation can now be solved e.g. with the pq-formula.

    $z_{1,2} = \frac{p}{2} \pm\sqrt{(\frac{p}{2})^2-q}$

    $z_{1,2} = \frac32 \pm\sqrt{(\frac32)^2-2}$
    $z_{1,2} = \frac32 \pm\sqrt{\frac14}$
    $z_{1,2} = \frac32 \pm\frac12$

    $z_1=2$ 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:


    Use both z-values.