Solve x^4-5x^2=6

$${ x }^{ 4 } -5 { x }^{ 2 } =6$$

Solve for x (complex solution)

$$x=\sqrt{6}\approx 2.449489743$$

$$x=-\sqrt{6}\approx -2.449489743$$

$$x=-i$$

$$x=i$$

Steps Using the Quadratic Formula

Subtract 6 from both sides.

$$x^{4}-5x^{2}-6=0$$

Substitute t for x^{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.

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

Do the calculations.

$$t=\frac{5±7}{2}$$

Solve the equation t=\frac{5±7}{2} when ± is plus and when ± is minus.

$$t=6 t=-1$$

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

$$x=-\sqrt{6} x=\sqrt{6} x=-i x=i$$

Solve for x

$$x=-\sqrt{6}\approx -2.449489743$$

$$x=\sqrt{6}\approx 2.449489743$$

Steps Using the Quadratic Formula

Subtract 6 from both sides.

$$x^{4}-5x^{2}-6=0$$

Substitute t for x^{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.

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

Do the calculations.

$$t=\frac{5±7}{2}$$

Solve the equation t=\frac{5±7}{2} when ± is plus and when ± is minus.

$$t=6 t=-1$$

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

$$x=\sqrt{6} x=-\sqrt{6}$$