Solve x^2+32^2={left(5/right)}^25/3x^2

$${ x }^{ 2 } + { 32 }^{ 2 } = { \left( \frac{ 5 }{ } \right) }^{ 2 } \frac{ 5 }{ 3 } { x }^{ 2 }$$

Solve for x

$$x = \frac{16 \sqrt{366}}{61} \approx 5.018000385$$

$$x = -\frac{16 \sqrt{366}}{61} \approx -5.018000385$$

Steps by Finding Square Root

Calculate 32 to the power of 2 and get 1024.

$$x^{2}+1024=\left(\frac{5}{1}\right)^{2}\times \frac{5}{3}x^{2}$$

Anything divided by one gives itself.

$$x^{2}+1024=5^{2}\times \frac{5}{3}x^{2}$$

Calculate 5 to the power of 2 and get 25.

$$x^{2}+1024=25\times \frac{5}{3}x^{2}$$

Multiply 25 and \frac{5}{3} to get \frac{125}{3}.

$$x^{2}+1024=\frac{125}{3}x^{2}$$

Subtract \frac{125}{3}x^{2} from both sides.

$$x^{2}+1024-\frac{125}{3}x^{2}=0$$

Combine x^{2} and -\frac{125}{3}x^{2} to get -\frac{122}{3}x^{2}.

$$-\frac{122}{3}x^{2}+1024=0$$

Subtract 1024 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{122}{3}x^{2}=-1024$$

Multiply both sides by -\frac{3}{122}, the reciprocal of -\frac{122}{3}.

$$x^{2}=-1024\left(-\frac{3}{122}\right)$$

Multiply -1024 and -\frac{3}{122} to get \frac{1536}{61}.

$$x^{2}=\frac{1536}{61}$$

Take the square root of both sides of the equation.

$$x=\frac{16\sqrt{366}}{61} x=-\frac{16\sqrt{366}}{61}$$

Steps Using the Quadratic Formula

Calculate 32 to the power of 2 and get 1024.

$$x^{2}+1024=\left(\frac{5}{1}\right)^{2}\times \frac{5}{3}x^{2}$$

Anything divided by one gives itself.

$$x^{2}+1024=5^{2}\times \frac{5}{3}x^{2}$$

Calculate 5 to the power of 2 and get 25.

$$x^{2}+1024=25\times \frac{5}{3}x^{2}$$

Multiply 25 and \frac{5}{3} to get \frac{125}{3}.

$$x^{2}+1024=\frac{125}{3}x^{2}$$

Subtract \frac{125}{3}x^{2} from both sides.

$$x^{2}+1024-\frac{125}{3}x^{2}=0$$

Combine x^{2} and -\frac{125}{3}x^{2} to get -\frac{122}{3}x^{2}.

$$-\frac{122}{3}x^{2}+1024=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{122}{3} for a, 0 for b, and 1024 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{122}{3}\right)\times 1024}}{2\left(-\frac{122}{3}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{122}{3}\right)\times 1024}}{2\left(-\frac{122}{3}\right)}$$

Multiply -4 times -\frac{122}{3}.

$$x=\frac{0±\sqrt{\frac{488}{3}\times 1024}}{2\left(-\frac{122}{3}\right)}$$

Multiply \frac{488}{3} times 1024.

$$x=\frac{0±\sqrt{\frac{499712}{3}}}{2\left(-\frac{122}{3}\right)}$$

Take the square root of \frac{499712}{3}.

$$x=\frac{0±\frac{64\sqrt{366}}{3}}{2\left(-\frac{122}{3}\right)}$$

Multiply 2 times -\frac{122}{3}.

$$x=\frac{0±\frac{64\sqrt{366}}{3}}{-\frac{244}{3}}$$

Now solve the equation x=\frac{0±\frac{64\sqrt{366}}{3}}{-\frac{244}{3}} when ± is plus.

$$x=-\frac{16\sqrt{366}}{61}$$

Now solve the equation x=\frac{0±\frac{64\sqrt{366}}{3}}{-\frac{244}{3}} when ± is minus.

$$x=\frac{16\sqrt{366}}{61}$$

The equation is now solved.

$$x=-\frac{16\sqrt{366}}{61} x=\frac{16\sqrt{366}}{61}$$