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$\fraction{\exponential{(5)}{2} + 15}{\exponential{(5)}{2}} = \fraction{\exponential{(2 r)}{2}}{\exponential{(5)}{2} + 15} $
Solve for r
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\frac{25+15}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Calculate 5 to the power of 2 and get 25.
\frac{40}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Add 25 and 15 to get 40.
\frac{40}{25}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Calculate 5 to the power of 2 and get 25.
\frac{8}{5}=\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{2^{2}r^{2}}{5^{2}+15}
Expand \left(2r\right)^{2}.
\frac{8}{5}=\frac{4r^{2}}{5^{2}+15}
Calculate 2 to the power of 2 and get 4.
\frac{8}{5}=\frac{4r^{2}}{25+15}
Calculate 5 to the power of 2 and get 25.
\frac{8}{5}=\frac{4r^{2}}{40}
Add 25 and 15 to get 40.
\frac{8}{5}=\frac{1}{10}r^{2}
Divide 4r^{2} by 40 to get \frac{1}{10}r^{2}.
\frac{1}{10}r^{2}=\frac{8}{5}
Swap sides so that all variable terms are on the left hand side.
\frac{1}{10}r^{2}-\frac{8}{5}=0
Subtract \frac{8}{5} from both sides.
r^{2}-16=0
Multiply both sides by 10.
\left(r-4\right)\left(r+4\right)=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).
r=4 r=-4
To find equation solutions, solve r-4=0 and r+4=0.
\frac{25+15}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Calculate 5 to the power of 2 and get 25.
\frac{40}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Add 25 and 15 to get 40.
\frac{40}{25}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Calculate 5 to the power of 2 and get 25.
\frac{8}{5}=\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{2^{2}r^{2}}{5^{2}+15}
Expand \left(2r\right)^{2}.
\frac{8}{5}=\frac{4r^{2}}{5^{2}+15}
Calculate 2 to the power of 2 and get 4.
\frac{8}{5}=\frac{4r^{2}}{25+15}
Calculate 5 to the power of 2 and get 25.
\frac{8}{5}=\frac{4r^{2}}{40}
Add 25 and 15 to get 40.
\frac{8}{5}=\frac{1}{10}r^{2}
Divide 4r^{2} by 40 to get \frac{1}{10}r^{2}.
\frac{1}{10}r^{2}=\frac{8}{5}
Swap sides so that all variable terms are on the left hand side.
r^{2}=\frac{8}{5}\times 10
Multiply both sides by 10, the reciprocal of \frac{1}{10}.
r^{2}=16
Multiply \frac{8}{5} and 10 to get 16.
r=4 r=-4
Take the square root of both sides of the equation.
\frac{25+15}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Calculate 5 to the power of 2 and get 25.
\frac{40}{5^{2}}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Add 25 and 15 to get 40.
\frac{40}{25}=\frac{\left(2r\right)^{2}}{5^{2}+15}
Calculate 5 to the power of 2 and get 25.
\frac{8}{5}=\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{2^{2}r^{2}}{5^{2}+15}
Expand \left(2r\right)^{2}.
\frac{8}{5}=\frac{4r^{2}}{5^{2}+15}
Calculate 2 to the power of 2 and get 4.
\frac{8}{5}=\frac{4r^{2}}{25+15}
Calculate 5 to the power of 2 and get 25.
\frac{8}{5}=\frac{4r^{2}}{40}
Add 25 and 15 to get 40.
\frac{8}{5}=\frac{1}{10}r^{2}
Divide 4r^{2} by 40 to get \frac{1}{10}r^{2}.
\frac{1}{10}r^{2}=\frac{8}{5}
Swap sides so that all variable terms are on the left hand side.
\frac{1}{10}r^{2}-\frac{8}{5}=0
Subtract \frac{8}{5} from both sides.
r=\frac{0±\sqrt{0^{2}-4\times \frac{1}{10}\left(-\frac{8}{5}\right)}}{2\times \frac{1}{10}}
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{-4\times \frac{1}{10}\left(-\frac{8}{5}\right)}}{2\times \frac{1}{10}}
Square 0.
r=\frac{0±\sqrt{-\frac{2}{5}\left(-\frac{8}{5}\right)}}{2\times \frac{1}{10}}
Multiply -4 times \frac{1}{10}.
r=\frac{0±\sqrt{\frac{16}{25}}}{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±\frac{4}{5}}{2\times \frac{1}{10}}
Take the square root of \frac{16}{25}.
r=\frac{0±\frac{4}{5}}{\frac{1}{5}}
Multiply 2 times \frac{1}{10}.
r=4
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 r=-4
The equation is now solved.