Solve x^2+2{left(3/5{x}^frac{2{1}}-3right)}^2-1=0

$${ x }^{ 2 } +2 { \left( \frac{ 3 }{ 5 } { x }^{ \frac{ 2 }{ 1 } } -3 \right) }^{ 2 } -1 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt[4]{30600}e^{\frac{-\arctan(\frac{\sqrt{263}}{31})i+2\pi i}{2}}}{6}\approx -2.14064034+0.526104088i$$

$$x=\frac{\sqrt[4]{30600}e^{-\frac{\arctan(\frac{\sqrt{263}}{31})i}{2}}}{6}\approx 2.14064034-0.526104088i$$

$$x=\frac{\sqrt[4]{30600}e^{\frac{\arctan(\frac{\sqrt{263}}{31})i+2\pi i}{2}}}{6}\approx -2.14064034-0.526104088i$$

$$x=\frac{\sqrt[4]{30600}e^{\frac{\arctan(\frac{\sqrt{263}}{31})i}{2}}}{6}\approx 2.14064034+0.526104088i$$

Steps Using the Quadratic Formula

Anything divided by one gives itself.

$$x^{2}+2\left(\frac{3}{5}x^{2}-3\right)^{2}-1=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{5}x^{2}-3\right)^{2}.

$$x^{2}+2\left(\frac{9}{25}\left(x^{2}\right)^{2}-\frac{18}{5}x^{2}+9\right)-1=0$$

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

$$x^{2}+2\left(\frac{9}{25}x^{4}-\frac{18}{5}x^{2}+9\right)-1=0$$

Use the distributive property to multiply 2 by \frac{9}{25}x^{4}-\frac{18}{5}x^{2}+9.

$$x^{2}+\frac{18}{25}x^{4}-\frac{36}{5}x^{2}+18-1=0$$

Combine x^{2} and -\frac{36}{5}x^{2} to get -\frac{31}{5}x^{2}.

$$-\frac{31}{5}x^{2}+\frac{18}{25}x^{4}+18-1=0$$

Subtract 1 from 18 to get 17.

$$-\frac{31}{5}x^{2}+\frac{18}{25}x^{4}+17=0$$

Substitute t for x^{2}.

$$\frac{18}{25}t^{2}-\frac{31}{5}t+17=0$$

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 \frac{18}{25} for a, -\frac{31}{5} for b, and 17 for c in the quadratic formula.

$$t=\frac{-\left(-\frac{31}{5}\right)±\sqrt{\left(-\frac{31}{5}\right)^{2}-4\times \frac{18}{25}\times 17}}{\frac{18}{25}\times 2}$$

Do the calculations.

$$t=\frac{\frac{31}{5}±\sqrt{-\frac{263}{25}}}{\frac{36}{25}}$$

Solve the equation t=\frac{\frac{31}{5}±\sqrt{-\frac{263}{25}}}{\frac{36}{25}} when ± is plus and when ± is minus.

$$t=\frac{155+5\sqrt{263}i}{36} t=\frac{-5\sqrt{263}i+155}{36}$$

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

$$x=\frac{\sqrt[4]{30600}e^{\frac{\arctan(\frac{\sqrt{263}}{31})i+2\pi i}{2}}}{6} x=\frac{\sqrt[4]{30600}e^{\frac{\arctan(\frac{\sqrt{263}}{31})i}{2}}}{6} x=\frac{\sqrt[4]{30600}e^{-\frac{\arctan(\frac{\sqrt{263}}{31})i}{2}}}{6} x=\frac{\sqrt[4]{30600}e^{\frac{-\arctan(\frac{\sqrt{263}}{31})i+2\pi i}{2}}}{6}$$