Solve 64/25+x^2=(sqrt{16+x^2}-8/5

$$\frac{ 64 }{ 25 } + { x }^{ 2 } = ( \sqrt{ 16+ { x }^{ 2 } } - \frac{ 8 }{ 5 }$$

Solve for x (complex solution)

$$x=-\frac{i\sqrt{366-10\sqrt{1209}}}{10}\approx -0-0.427705822i$$

$$x=\frac{i\sqrt{366-10\sqrt{1209}}}{10}\approx 0.427705822i$$

Solution Steps

Subtract -\frac{8}{5} from both sides of the equation.

$$\frac{64}{25}+x^{2}+\frac{8}{5}=\sqrt{16+x^{2}}$$

Add \frac{64}{25} and \frac{8}{5} to get \frac{104}{25}.

$$\frac{104}{25}+x^{2}=\sqrt{16+x^{2}}$$

Square both sides of the equation.

$$\left(\frac{104}{25}+x^{2}\right)^{2}=\left(\sqrt{16+x^{2}}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{104}{25}+x^{2}\right)^{2}.

$$\frac{10816}{625}+\frac{208}{25}x^{2}+\left(x^{2}\right)^{2}=\left(\sqrt{16+x^{2}}\right)^{2}$$

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

$$\frac{10816}{625}+\frac{208}{25}x^{2}+x^{4}=\left(\sqrt{16+x^{2}}\right)^{2}$$

Calculate \sqrt{16+x^{2}} to the power of 2 and get 16+x^{2}.

$$\frac{10816}{625}+\frac{208}{25}x^{2}+x^{4}=16+x^{2}$$

Subtract 16 from both sides.

$$\frac{10816}{625}+\frac{208}{25}x^{2}+x^{4}-16=x^{2}$$

Subtract 16 from \frac{10816}{625} to get \frac{816}{625}.

$$\frac{816}{625}+\frac{208}{25}x^{2}+x^{4}=x^{2}$$

Subtract x^{2} from both sides.

$$\frac{816}{625}+\frac{208}{25}x^{2}+x^{4}-x^{2}=0$$

Combine \frac{208}{25}x^{2} and -x^{2} to get \frac{183}{25}x^{2}.

$$\frac{816}{625}+\frac{183}{25}x^{2}+x^{4}=0$$

Substitute t for x^{2}.

$$t^{2}+\frac{183}{25}t+\frac{816}{625}=0$$

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, \frac{183}{25} for b, and \frac{816}{625} for c in the quadratic formula.

$$t=\frac{-\frac{183}{25}±\sqrt{\left(\frac{183}{25}\right)^{2}-4\times 1\times \frac{816}{625}}}{2}$$

Do the calculations.

$$t=\frac{-\frac{183}{25}±\frac{1}{5}\sqrt{1209}}{2}$$

Solve the equation t=\frac{-\frac{183}{25}±\frac{1}{5}\sqrt{1209}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{1209}}{10}-\frac{183}{50} t=-\frac{\sqrt{1209}}{10}-\frac{183}{50}$$

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

$$x=-i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}} x=i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}} x=-i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}} x=i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}}$$

Substitute -i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}} for x in the equation \frac{64}{25}+x^{2}=\sqrt{16+x^{2}}-\frac{8}{5}.

$$\frac{64}{25}+\left(-i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}}\right)^{2}=\sqrt{16+\left(-i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}}\right)^{2}}-\frac{8}{5}$$

Simplify. The value x=-i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}} satisfies the equation.

$$-\frac{11}{10}+\frac{1}{10}\times 1209^{\frac{1}{2}}=-\frac{11}{10}+\frac{1}{10}\times 1209^{\frac{1}{2}}$$

Substitute i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}} for x in the equation \frac{64}{25}+x^{2}=\sqrt{16+x^{2}}-\frac{8}{5}.

$$\frac{64}{25}+\left(i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}}\right)^{2}=\sqrt{16+\left(i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}}\right)^{2}}-\frac{8}{5}$$

Simplify. The value x=i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}} satisfies the equation.

$$-\frac{11}{10}+\frac{1}{10}\times 1209^{\frac{1}{2}}=-\frac{11}{10}+\frac{1}{10}\times 1209^{\frac{1}{2}}$$

Substitute -i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}} for x in the equation \frac{64}{25}+x^{2}=\sqrt{16+x^{2}}-\frac{8}{5}.

$$\frac{64}{25}+\left(-i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}}\right)^{2}=\sqrt{16+\left(-i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}}\right)^{2}}-\frac{8}{5}$$

Simplify. The value x=-i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{11}{10}-\frac{1}{10}\times 1209^{\frac{1}{2}}=-\frac{21}{10}+\frac{1}{10}\times 1209^{\frac{1}{2}}$$

Substitute i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}} for x in the equation \frac{64}{25}+x^{2}=\sqrt{16+x^{2}}-\frac{8}{5}.

$$\frac{64}{25}+\left(i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}}\right)^{2}=\sqrt{16+\left(i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}}\right)^{2}}-\frac{8}{5}$$

Simplify. The value x=i\sqrt{\frac{\sqrt{1209}}{10}+\frac{183}{50}} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{11}{10}-\frac{1}{10}\times 1209^{\frac{1}{2}}=-\frac{21}{10}+\frac{1}{10}\times 1209^{\frac{1}{2}}$$

List all solutions of x^{2}+\frac{104}{25}=\sqrt{x^{2}+16}.

$$x=-i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}} x=i\sqrt{-\frac{\sqrt{1209}}{10}+\frac{183}{50}}$$