Solve {left(27sqrt{2}right)}^2-{left(frac{78sqrt{10}}{125}right)}^2=x^2

$${ \left(27 \sqrt{ 2 } \right) }^{ 2 } - { \left( \frac{ 78 \sqrt{ 10 } }{ 125 } \right) }^{ 2 } = { x }^{ 2 }$$

Solve for x

$$x = \frac{3 \sqrt{2524490}}{125} \approx 38.132744984$$

$$x = -\frac{3 \sqrt{2524490}}{125} \approx -38.132744984$$

Steps by Finding Square Root

Expand \left(27\sqrt{2}\right)^{2}.

$$27^{2}\left(\sqrt{2}\right)^{2}-\left(\frac{78\sqrt{10}}{125}\right)^{2}=x^{2}$$

Calculate 27 to the power of 2 and get 729.

$$729\left(\sqrt{2}\right)^{2}-\left(\frac{78\sqrt{10}}{125}\right)^{2}=x^{2}$$

The square of \sqrt{2} is 2.

$$729\times 2-\left(\frac{78\sqrt{10}}{125}\right)^{2}=x^{2}$$

Multiply 729 and 2 to get 1458.

$$1458-\left(\frac{78\sqrt{10}}{125}\right)^{2}=x^{2}$$

To raise \frac{78\sqrt{10}}{125} to a power, raise both numerator and denominator to the power and then divide.

$$1458-\frac{\left(78\sqrt{10}\right)^{2}}{125^{2}}=x^{2}$$

Expand \left(78\sqrt{10}\right)^{2}.

$$1458-\frac{78^{2}\left(\sqrt{10}\right)^{2}}{125^{2}}=x^{2}$$

Calculate 78 to the power of 2 and get 6084.

$$1458-\frac{6084\left(\sqrt{10}\right)^{2}}{125^{2}}=x^{2}$$

The square of \sqrt{10} is 10.

$$1458-\frac{6084\times 10}{125^{2}}=x^{2}$$

Multiply 6084 and 10 to get 60840.

$$1458-\frac{60840}{125^{2}}=x^{2}$$

Calculate 125 to the power of 2 and get 15625.

$$1458-\frac{60840}{15625}=x^{2}$$

Reduce the fraction \frac{60840}{15625} to lowest terms by extracting and canceling out 5.

$$1458-\frac{12168}{3125}=x^{2}$$

Subtract \frac{12168}{3125} from 1458 to get \frac{4544082}{3125}.

$$\frac{4544082}{3125}=x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}=\frac{4544082}{3125}$$

Take the square root of both sides of the equation.

$$x=\frac{3\sqrt{2524490}}{125} x=-\frac{3\sqrt{2524490}}{125}$$

Steps Using the Quadratic Formula

Expand \left(27\sqrt{2}\right)^{2}.

$$27^{2}\left(\sqrt{2}\right)^{2}-\left(\frac{78\sqrt{10}}{125}\right)^{2}=x^{2}$$

Calculate 27 to the power of 2 and get 729.

$$729\left(\sqrt{2}\right)^{2}-\left(\frac{78\sqrt{10}}{125}\right)^{2}=x^{2}$$

The square of \sqrt{2} is 2.

$$729\times 2-\left(\frac{78\sqrt{10}}{125}\right)^{2}=x^{2}$$

Multiply 729 and 2 to get 1458.

$$1458-\left(\frac{78\sqrt{10}}{125}\right)^{2}=x^{2}$$

To raise \frac{78\sqrt{10}}{125} to a power, raise both numerator and denominator to the power and then divide.

$$1458-\frac{\left(78\sqrt{10}\right)^{2}}{125^{2}}=x^{2}$$

Expand \left(78\sqrt{10}\right)^{2}.

$$1458-\frac{78^{2}\left(\sqrt{10}\right)^{2}}{125^{2}}=x^{2}$$

Calculate 78 to the power of 2 and get 6084.

$$1458-\frac{6084\left(\sqrt{10}\right)^{2}}{125^{2}}=x^{2}$$

The square of \sqrt{10} is 10.

$$1458-\frac{6084\times 10}{125^{2}}=x^{2}$$

Multiply 6084 and 10 to get 60840.

$$1458-\frac{60840}{125^{2}}=x^{2}$$

Calculate 125 to the power of 2 and get 15625.

$$1458-\frac{60840}{15625}=x^{2}$$

Reduce the fraction \frac{60840}{15625} to lowest terms by extracting and canceling out 5.

$$1458-\frac{12168}{3125}=x^{2}$$

Subtract \frac{12168}{3125} from 1458 to get \frac{4544082}{3125}.

$$\frac{4544082}{3125}=x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}=\frac{4544082}{3125}$$

Subtract \frac{4544082}{3125} from both sides.

$$x^{2}-\frac{4544082}{3125}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{4544082}{3125} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{4544082}{3125}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{4544082}{3125}\right)}}{2}$$

Multiply -4 times -\frac{4544082}{3125}.

$$x=\frac{0±\sqrt{\frac{18176328}{3125}}}{2}$$

Take the square root of \frac{18176328}{3125}.

$$x=\frac{0±\frac{6\sqrt{2524490}}{125}}{2}$$

Now solve the equation x=\frac{0±\frac{6\sqrt{2524490}}{125}}{2} when ± is plus.

$$x=\frac{3\sqrt{2524490}}{125}$$

Now solve the equation x=\frac{0±\frac{6\sqrt{2524490}}{125}}{2} when ± is minus.

$$x=-\frac{3\sqrt{2524490}}{125}$$

The equation is now solved.

$$x=\frac{3\sqrt{2524490}}{125} x=-\frac{3\sqrt{2524490}}{125}$$