A biquadratic equation (in a narrower sense) is the special case of the quartic equation containing no odd degree terms:

(1)

For solving a biquadratic equation (1) one does not need the quartic formula since the equation may be thought a quadratic equation with respect to $x^2$ , i.e. $$a(x^2)^2+bx^2+c = 0,$$ whence $$x^2 = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ (see quadratic formula or quadratic equation in $\mathbb{C}$ ). Taking square roots of the values of $x^2$ (see taking square root algebraically), one obtains the four roots of (1).

Example. Solve the biquadratic equation

(2)

(3)

(4)

1) $x = 1+\sqrt2+\sqrt3$
$(x-1)^2 = 2+2\sqrt{6}+3$
$y^2-5 = 2\sqrt{6}$
$y^4-10y^2+1 = 0$ (one has substituted $x-1 := y$ )

2) $x = \sqrt{\sqrt{2}-1}$
$x^2 = \sqrt{2}-1$
$(x^2+1)^2 = 2$
$x^4+2x^2-1 = 0$