Solve 13^2-x^2=(2sqrt{61})^2-(2x)^2

$$13 ^ { 2 } - x ^ { 2 } = ( 2 \sqrt { 61 } ) ^ { 2 } - ( 2 x ) ^ { 2 }$$

Solve for x

$$x=-5$$

$$x=5$$

Steps Using Difference of Squares

Calculate 13 to the power of 2 and get 169.

$$169-x^{2}=\left(2\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

Expand \left(2\sqrt{61}\right)^{2}.

$$169-x^{2}=2^{2}\left(\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$169-x^{2}=4\left(\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

The square of \sqrt{61} is 61.

$$169-x^{2}=4\times 61-\left(2x\right)^{2}$$

Multiply 4 and 61 to get 244.

$$169-x^{2}=244-\left(2x\right)^{2}$$

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

$$169-x^{2}=244-2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$169-x^{2}=244-4x^{2}$$

Subtract 244 from both sides.

$$169-x^{2}-244=-4x^{2}$$

Subtract 244 from 169 to get -75.

$$-75-x^{2}=-4x^{2}$$

Add 4x^{2} to both sides.

$$-75-x^{2}+4x^{2}=0$$

Combine -x^{2} and 4x^{2} to get 3x^{2}.

$$-75+3x^{2}=0$$

Divide both sides by 3.

$$-25+x^{2}=0$$

Consider -25+x^{2}. Rewrite -25+x^{2} as x^{2}-5^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-5\right)\left(x+5\right)=0$$

To find equation solutions, solve x-5=0 and x+5=0.

$$x=5 x=-5$$

Steps by Finding Square Root

Calculate 13 to the power of 2 and get 169.

$$169-x^{2}=\left(2\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

Expand \left(2\sqrt{61}\right)^{2}.

$$169-x^{2}=2^{2}\left(\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$169-x^{2}=4\left(\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

The square of \sqrt{61} is 61.

$$169-x^{2}=4\times 61-\left(2x\right)^{2}$$

Multiply 4 and 61 to get 244.

$$169-x^{2}=244-\left(2x\right)^{2}$$

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

$$169-x^{2}=244-2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$169-x^{2}=244-4x^{2}$$

Add 4x^{2} to both sides.

$$169-x^{2}+4x^{2}=244$$

Combine -x^{2} and 4x^{2} to get 3x^{2}.

$$169+3x^{2}=244$$

Subtract 169 from both sides.

$$3x^{2}=244-169$$

Subtract 169 from 244 to get 75.

$$3x^{2}=75$$

Divide both sides by 3.

$$x^{2}=\frac{75}{3}$$

Divide 75 by 3 to get 25.

$$x^{2}=25$$

Take the square root of both sides of the equation.

$$x=5 x=-5$$

Steps Using the Quadratic Formula

Calculate 13 to the power of 2 and get 169.

$$169-x^{2}=\left(2\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

Expand \left(2\sqrt{61}\right)^{2}.

$$169-x^{2}=2^{2}\left(\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$169-x^{2}=4\left(\sqrt{61}\right)^{2}-\left(2x\right)^{2}$$

The square of \sqrt{61} is 61.

$$169-x^{2}=4\times 61-\left(2x\right)^{2}$$

Multiply 4 and 61 to get 244.

$$169-x^{2}=244-\left(2x\right)^{2}$$

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

$$169-x^{2}=244-2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$169-x^{2}=244-4x^{2}$$

Subtract 244 from both sides.

$$169-x^{2}-244=-4x^{2}$$

Subtract 244 from 169 to get -75.

$$-75-x^{2}=-4x^{2}$$

Add 4x^{2} to both sides.

$$-75-x^{2}+4x^{2}=0$$

Combine -x^{2} and 4x^{2} to get 3x^{2}.

$$-75+3x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$3x^{2}-75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(-75\right)}}{2\times 3}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 3\left(-75\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{0±\sqrt{-12\left(-75\right)}}{2\times 3}$$

Multiply -12 times -75.

$$x=\frac{0±\sqrt{900}}{2\times 3}$$

Take the square root of 900.

$$x=\frac{0±30}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{0±30}{6}$$

Now solve the equation x=\frac{0±30}{6} when ± is plus. Divide 30 by 6.

$$x=5$$

Now solve the equation x=\frac{0±30}{6} when ± is minus. Divide -30 by 6.

$$x=-5$$

The equation is now solved.

$$x=5 x=-5$$