Solve z^4-5z^2+6=0

$${ z }^{ 4 } -5 { z }^{ 2 } +6 = 0$$

Solve for z

$$z=-\sqrt{3}\approx -1.732050808$$

$$z=\sqrt{3}\approx 1.732050808$$

$$z=\sqrt{2}\approx 1.414213562$$

$$z=-\sqrt{2}\approx -1.414213562$$

Steps Using the Quadratic Formula

Substitute t for z^{2}.

$$t^{2}-5t+6=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -5 for b, and 6 for c in the quadratic formula.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\times 6}}{2}$$

Do the calculations.

$$t=\frac{5±1}{2}$$

Solve the equation t=\frac{5±1}{2} when ± is plus and when ± is minus.

$$t=3 t=2$$

Since z=t^{2}, the solutions are obtained by evaluating z=±\sqrt{t} for each t.

$$z=\sqrt{3} z=-\sqrt{3} z=\sqrt{2} z=-\sqrt{2}$$