Solve (4x^2)^2+91x^2-225=0

$$( 4 x ^ { 2 } ) ^ { 2 } + 91 x ^ { 2 } - 225 = 0$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{2\left(\sqrt{22681}+91\right)}}{8}\approx 2.747738416i$$

$$x=-\frac{i\sqrt{2\left(\sqrt{22681}+91\right)}}{8}\approx -0-2.747738416i$$

$$x = -\frac{\sqrt{2 {(\sqrt{22681} - 91)}}}{8} \approx -1.364758733$$

$$x = \frac{\sqrt{2 {(\sqrt{22681} - 91)}}}{8} \approx 1.364758733$$

Steps Using the Quadratic Formula

Expand \left(4x^{2}\right)^{2}.

$$4^{2}\left(x^{2}\right)^{2}+91x^{2}-225=0$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4^{2}x^{4}+91x^{2}-225=0$$

Calculate 4 to the power of 2 and get 16.

$$16x^{4}+91x^{2}-225=0$$

Substitute t for x^{2}.

$$16t^{2}+91t-225=0$$

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 16 for a, 91 for b, and -225 for c in the quadratic formula.

$$t=\frac{-91±\sqrt{91^{2}-4\times 16\left(-225\right)}}{2\times 16}$$

Do the calculations.

$$t=\frac{-91±\sqrt{22681}}{32}$$

Solve the equation t=\frac{-91±\sqrt{22681}}{32} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{22681}-91}{32} t=\frac{-\sqrt{22681}-91}{32}$$

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

$$x=-\sqrt{\frac{\sqrt{22681}-91}{32}} x=\sqrt{\frac{\sqrt{22681}-91}{32}} x=-i\sqrt{\frac{\sqrt{22681}+91}{32}} x=i\sqrt{\frac{\sqrt{22681}+91}{32}}$$

Solve for x

$$x = -\frac{\sqrt{2 {(\sqrt{22681} - 91)}}}{8} \approx -1.364758733$$

$$x = \frac{\sqrt{2 {(\sqrt{22681} - 91)}}}{8} \approx 1.364758733$$

Steps Using the Quadratic Formula

Expand \left(4x^{2}\right)^{2}.

$$4^{2}\left(x^{2}\right)^{2}+91x^{2}-225=0$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4^{2}x^{4}+91x^{2}-225=0$$

Calculate 4 to the power of 2 and get 16.

$$16x^{4}+91x^{2}-225=0$$

Substitute t for x^{2}.

$$16t^{2}+91t-225=0$$

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 16 for a, 91 for b, and -225 for c in the quadratic formula.

$$t=\frac{-91±\sqrt{91^{2}-4\times 16\left(-225\right)}}{2\times 16}$$

Do the calculations.

$$t=\frac{-91±\sqrt{22681}}{32}$$

Solve the equation t=\frac{-91±\sqrt{22681}}{32} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{22681}-91}{32} t=\frac{-\sqrt{22681}-91}{32}$$

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

$$x=\frac{\sqrt{\frac{\sqrt{22681}-91}{8}}}{2} x=-\frac{\sqrt{\frac{\sqrt{22681}-91}{8}}}{2}$$