Solve 5^2+15/5^2=left(2rright)^2/5^2+15

$$\frac{ { 5 }^{ 2 } +15 }{ { 5 }^{ 2 } } = \frac{ { \left(2r \right) }^{ 2 } }{ { 5 }^{ 2 } +15 }$$

Solve for r

$$r=4$$

$$r=-4$$

Steps Using Difference of Squares

Calculate 5 to the power of 2 and get 25.

$$\frac{25+15}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Add 25 and 15 to get 40.

$$\frac{40}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{40}{25}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Reduce the fraction \frac{40}{25} to lowest terms by extracting and canceling out 5.

$$\frac{8}{5}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

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

$$\frac{8}{5}=\frac{2^{2}r^{2}}{5^{2}+15}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{8}{5}=\frac{4r^{2}}{5^{2}+15}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{8}{5}=\frac{4r^{2}}{25+15}$$

Add 25 and 15 to get 40.

$$\frac{8}{5}=\frac{4r^{2}}{40}$$

Divide 4r^{2} by 40 to get \frac{1}{10}r^{2}.

$$\frac{8}{5}=\frac{1}{10}r^{2}$$

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

$$\frac{1}{10}r^{2}=\frac{8}{5}$$

Subtract \frac{8}{5} from both sides.

$$\frac{1}{10}r^{2}-\frac{8}{5}=0$$

Multiply both sides by 10.

$$r^{2}-16=0$$

Consider r^{2}-16. Rewrite r^{2}-16 as r^{2}-4^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(r-4\right)\left(r+4\right)=0$$

To find equation solutions, solve r-4=0 and r+4=0.

$$r=4 r=-4$$

Steps by Finding Square Root

Calculate 5 to the power of 2 and get 25.

$$\frac{25+15}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Add 25 and 15 to get 40.

$$\frac{40}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{40}{25}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Reduce the fraction \frac{40}{25} to lowest terms by extracting and canceling out 5.

$$\frac{8}{5}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

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

$$\frac{8}{5}=\frac{2^{2}r^{2}}{5^{2}+15}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{8}{5}=\frac{4r^{2}}{5^{2}+15}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{8}{5}=\frac{4r^{2}}{25+15}$$

Add 25 and 15 to get 40.

$$\frac{8}{5}=\frac{4r^{2}}{40}$$

Divide 4r^{2} by 40 to get \frac{1}{10}r^{2}.

$$\frac{8}{5}=\frac{1}{10}r^{2}$$

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

$$\frac{1}{10}r^{2}=\frac{8}{5}$$

Multiply both sides by 10, the reciprocal of \frac{1}{10}.

$$r^{2}=\frac{8}{5}\times 10$$

Multiply \frac{8}{5} and 10 to get 16.

$$r^{2}=16$$

Take the square root of both sides of the equation.

$$r=4 r=-4$$

Steps Using the Quadratic Formula

Calculate 5 to the power of 2 and get 25.

$$\frac{25+15}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Add 25 and 15 to get 40.

$$\frac{40}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{40}{25}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

Reduce the fraction \frac{40}{25} to lowest terms by extracting and canceling out 5.

$$\frac{8}{5}=\frac{\left(2r\right)^{2}}{5^{2}+15}$$

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

$$\frac{8}{5}=\frac{2^{2}r^{2}}{5^{2}+15}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{8}{5}=\frac{4r^{2}}{5^{2}+15}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{8}{5}=\frac{4r^{2}}{25+15}$$

Add 25 and 15 to get 40.

$$\frac{8}{5}=\frac{4r^{2}}{40}$$

Divide 4r^{2} by 40 to get \frac{1}{10}r^{2}.

$$\frac{8}{5}=\frac{1}{10}r^{2}$$

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

$$\frac{1}{10}r^{2}=\frac{8}{5}$$

Subtract \frac{8}{5} from both sides.

$$\frac{1}{10}r^{2}-\frac{8}{5}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{10} for a, 0 for b, and -\frac{8}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{0±\sqrt{0^{2}-4\times \frac{1}{10}\left(-\frac{8}{5}\right)}}{2\times \frac{1}{10}}$$

Square 0.

$$r=\frac{0±\sqrt{-4\times \frac{1}{10}\left(-\frac{8}{5}\right)}}{2\times \frac{1}{10}}$$

Multiply -4 times \frac{1}{10}.

$$r=\frac{0±\sqrt{-\frac{2}{5}\left(-\frac{8}{5}\right)}}{2\times \frac{1}{10}}$$

Multiply -\frac{2}{5} times -\frac{8}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$r=\frac{0±\sqrt{\frac{16}{25}}}{2\times \frac{1}{10}}$$

Take the square root of \frac{16}{25}.

$$r=\frac{0±\frac{4}{5}}{2\times \frac{1}{10}}$$

Multiply 2 times \frac{1}{10}.

$$r=\frac{0±\frac{4}{5}}{\frac{1}{5}}$$

Now solve the equation r=\frac{0±\frac{4}{5}}{\frac{1}{5}} when ± is plus.

$$r=4$$

Now solve the equation r=\frac{0±\frac{4}{5}}{\frac{1}{5}} when ± is minus.

$$r=-4$$

The equation is now solved.

$$r=4 r=-4$$