Solve 5832=5000(1+R/100)^2

$$5832 = 5000 ( 1 + \frac { R } { 100 } ) ^ { 2 }$$

Solve for R

$$R=8$$

$$R=-208$$

Steps Using the Quadratic Formula

Divide both sides by 5000.

$$\frac{5832}{5000}=\left(1+\frac{R}{100}\right)^{2}$$

Reduce the fraction \frac{5832}{5000} to lowest terms by extracting and canceling out 8.

$$\frac{729}{625}=\left(1+\frac{R}{100}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\frac{R}{100}\right)^{2}.

$$\frac{729}{625}=1+2\times \frac{R}{100}+\left(\frac{R}{100}\right)^{2}$$

Cancel out 100, the greatest common factor in 2 and 100.

$$\frac{729}{625}=1+\frac{R}{50}+\left(\frac{R}{100}\right)^{2}$$

To raise \frac{R}{100} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{729}{625}=1+\frac{R}{50}+\frac{R^{2}}{100^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{100^{2}}{100^{2}}.

$$\frac{729}{625}=\frac{100^{2}}{100^{2}}+\frac{R}{50}+\frac{R^{2}}{100^{2}}$$

Since \frac{100^{2}}{100^{2}} and \frac{R^{2}}{100^{2}} have the same denominator, add them by adding their numerators.

$$\frac{729}{625}=\frac{100^{2}+R^{2}}{100^{2}}+\frac{R}{50}$$

Combine like terms in 100^{2}+R^{2}.

$$\frac{729}{625}=\frac{10000+R^{2}}{100^{2}}+\frac{R}{50}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 100^{2} and 50 is 10000. Multiply \frac{R}{50} times \frac{200}{200}.

$$\frac{729}{625}=\frac{10000+R^{2}}{10000}+\frac{200R}{10000}$$

Since \frac{10000+R^{2}}{10000} and \frac{200R}{10000} have the same denominator, add them by adding their numerators.

$$\frac{729}{625}=\frac{10000+R^{2}+200R}{10000}$$

Divide each term of 10000+R^{2}+200R by 10000 to get 1+\frac{1}{10000}R^{2}+\frac{1}{50}R.

$$\frac{729}{625}=1+\frac{1}{10000}R^{2}+\frac{1}{50}R$$

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

$$1+\frac{1}{10000}R^{2}+\frac{1}{50}R=\frac{729}{625}$$

Subtract \frac{729}{625} from both sides.

$$1+\frac{1}{10000}R^{2}+\frac{1}{50}R-\frac{729}{625}=0$$

Subtract \frac{729}{625} from 1 to get -\frac{104}{625}.

$$-\frac{104}{625}+\frac{1}{10000}R^{2}+\frac{1}{50}R=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\frac{1}{10000}R^{2}+\frac{1}{50}R-\frac{104}{625}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{10000} for a, \frac{1}{50} for b, and -\frac{104}{625} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$R=\frac{-\frac{1}{50}±\sqrt{\left(\frac{1}{50}\right)^{2}-4\times \frac{1}{10000}\left(-\frac{104}{625}\right)}}{2\times \frac{1}{10000}}$$

Square \frac{1}{50} by squaring both the numerator and the denominator of the fraction.

$$R=\frac{-\frac{1}{50}±\sqrt{\frac{1}{2500}-4\times \frac{1}{10000}\left(-\frac{104}{625}\right)}}{2\times \frac{1}{10000}}$$

Multiply -4 times \frac{1}{10000}.

$$R=\frac{-\frac{1}{50}±\sqrt{\frac{1}{2500}-\frac{1}{2500}\left(-\frac{104}{625}\right)}}{2\times \frac{1}{10000}}$$

Multiply -\frac{1}{2500} times -\frac{104}{625} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$R=\frac{-\frac{1}{50}±\sqrt{\frac{1}{2500}+\frac{26}{390625}}}{2\times \frac{1}{10000}}$$

Add \frac{1}{2500} to \frac{26}{390625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$R=\frac{-\frac{1}{50}±\sqrt{\frac{729}{1562500}}}{2\times \frac{1}{10000}}$$

Take the square root of \frac{729}{1562500}.

$$R=\frac{-\frac{1}{50}±\frac{27}{1250}}{2\times \frac{1}{10000}}$$

Multiply 2 times \frac{1}{10000}.

$$R=\frac{-\frac{1}{50}±\frac{27}{1250}}{\frac{1}{5000}}$$

Now solve the equation R=\frac{-\frac{1}{50}±\frac{27}{1250}}{\frac{1}{5000}} when ± is plus. Add -\frac{1}{50} to \frac{27}{1250} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$R=\frac{\frac{1}{625}}{\frac{1}{5000}}$$

Divide \frac{1}{625} by \frac{1}{5000} by multiplying \frac{1}{625} by the reciprocal of \frac{1}{5000}.

$$R=8$$

Now solve the equation R=\frac{-\frac{1}{50}±\frac{27}{1250}}{\frac{1}{5000}} when ± is minus. Subtract \frac{27}{1250} from -\frac{1}{50} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$R=-\frac{\frac{26}{625}}{\frac{1}{5000}}$$

Divide -\frac{26}{625} by \frac{1}{5000} by multiplying -\frac{26}{625} by the reciprocal of \frac{1}{5000}.

$$R=-208$$

The equation is now solved.

$$R=8 R=-208$$

Steps for Completing the Square

Divide both sides by 5000.

$$\frac{5832}{5000}=\left(1+\frac{R}{100}\right)^{2}$$

Reduce the fraction \frac{5832}{5000} to lowest terms by extracting and canceling out 8.

$$\frac{729}{625}=\left(1+\frac{R}{100}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\frac{R}{100}\right)^{2}.

$$\frac{729}{625}=1+2\times \frac{R}{100}+\left(\frac{R}{100}\right)^{2}$$

Cancel out 100, the greatest common factor in 2 and 100.

$$\frac{729}{625}=1+\frac{R}{50}+\left(\frac{R}{100}\right)^{2}$$

To raise \frac{R}{100} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{729}{625}=1+\frac{R}{50}+\frac{R^{2}}{100^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{100^{2}}{100^{2}}.

$$\frac{729}{625}=\frac{100^{2}}{100^{2}}+\frac{R}{50}+\frac{R^{2}}{100^{2}}$$

Since \frac{100^{2}}{100^{2}} and \frac{R^{2}}{100^{2}} have the same denominator, add them by adding their numerators.

$$\frac{729}{625}=\frac{100^{2}+R^{2}}{100^{2}}+\frac{R}{50}$$

Combine like terms in 100^{2}+R^{2}.

$$\frac{729}{625}=\frac{10000+R^{2}}{100^{2}}+\frac{R}{50}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 100^{2} and 50 is 10000. Multiply \frac{R}{50} times \frac{200}{200}.

$$\frac{729}{625}=\frac{10000+R^{2}}{10000}+\frac{200R}{10000}$$

Since \frac{10000+R^{2}}{10000} and \frac{200R}{10000} have the same denominator, add them by adding their numerators.

$$\frac{729}{625}=\frac{10000+R^{2}+200R}{10000}$$

Divide each term of 10000+R^{2}+200R by 10000 to get 1+\frac{1}{10000}R^{2}+\frac{1}{50}R.

$$\frac{729}{625}=1+\frac{1}{10000}R^{2}+\frac{1}{50}R$$

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

$$1+\frac{1}{10000}R^{2}+\frac{1}{50}R=\frac{729}{625}$$

Subtract 1 from both sides.

$$\frac{1}{10000}R^{2}+\frac{1}{50}R=\frac{729}{625}-1$$

Subtract 1 from \frac{729}{625} to get \frac{104}{625}.

$$\frac{1}{10000}R^{2}+\frac{1}{50}R=\frac{104}{625}$$

Multiply both sides by 10000.

$$\frac{\frac{1}{10000}R^{2}+\frac{1}{50}R}{\frac{1}{10000}}=\frac{\frac{104}{625}}{\frac{1}{10000}}$$

Dividing by \frac{1}{10000} undoes the multiplication by \frac{1}{10000}.

$$R^{2}+\frac{\frac{1}{50}}{\frac{1}{10000}}R=\frac{\frac{104}{625}}{\frac{1}{10000}}$$

Divide \frac{1}{50} by \frac{1}{10000} by multiplying \frac{1}{50} by the reciprocal of \frac{1}{10000}.

$$R^{2}+200R=\frac{\frac{104}{625}}{\frac{1}{10000}}$$

Divide \frac{104}{625} by \frac{1}{10000} by multiplying \frac{104}{625} by the reciprocal of \frac{1}{10000}.

$$R^{2}+200R=1664$$

Divide 200, the coefficient of the x term, by 2 to get 100. Then add the square of 100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$R^{2}+200R+100^{2}=1664+100^{2}$$

Square 100.

$$R^{2}+200R+10000=1664+10000$$

Add 1664 to 10000.

$$R^{2}+200R+10000=11664$$

Factor R^{2}+200R+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(R+100\right)^{2}=11664$$

Take the square root of both sides of the equation.

$$\sqrt{\left(R+100\right)^{2}}=\sqrt{11664}$$

Simplify.

$$R+100=108 R+100=-108$$

Subtract 100 from both sides of the equation.

$$R=8 R=-208$$