Solve 2x^4-9=11x^2

$$2 x ^ { 4 } - 9 = 11 x ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{\sqrt{193}-11}}{2}\approx 0.850359334i$$

$$x=-\frac{i\sqrt{\sqrt{193}-11}}{2}\approx -0-0.850359334i$$

$$x = -\frac{\sqrt{\sqrt{193} + 11}}{2} \approx -2.494616403$$

$$x = \frac{\sqrt{\sqrt{193} + 11}}{2} \approx 2.494616403$$

Steps Using the Quadratic Formula

Subtract 11x^{2} from both sides.

$$2x^{4}-9-11x^{2}=0$$

Substitute t for x^{2}.

$$2t^{2}-11t-9=0$$

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 2 for a, -11 for b, and -9 for c in the quadratic formula.

$$t=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\left(-9\right)}}{2\times 2}$$

Do the calculations.

$$t=\frac{11±\sqrt{193}}{4}$$

Solve the equation t=\frac{11±\sqrt{193}}{4} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{193}+11}{4} t=\frac{11-\sqrt{193}}{4}$$

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

$$x=-\frac{\sqrt{\sqrt{193}+11}}{2} x=\frac{\sqrt{\sqrt{193}+11}}{2} x=-i\sqrt{-\frac{11-\sqrt{193}}{4}} x=i\sqrt{-\frac{11-\sqrt{193}}{4}}$$

Solve for x

$$x = -\frac{\sqrt{\sqrt{193} + 11}}{2} \approx -2.494616403$$

$$x = \frac{\sqrt{\sqrt{193} + 11}}{2} \approx 2.494616403$$

Steps Using the Quadratic Formula

Subtract 11x^{2} from both sides.

$$2x^{4}-9-11x^{2}=0$$

Substitute t for x^{2}.

$$2t^{2}-11t-9=0$$

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 2 for a, -11 for b, and -9 for c in the quadratic formula.

$$t=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\left(-9\right)}}{2\times 2}$$

Do the calculations.

$$t=\frac{11±\sqrt{193}}{4}$$

Solve the equation t=\frac{11±\sqrt{193}}{4} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{193}+11}{4} t=\frac{11-\sqrt{193}}{4}$$

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

$$x=\frac{\sqrt{\sqrt{193}+11}}{2} x=-\frac{\sqrt{\sqrt{193}+11}}{2}$$