Solve 3600/2500=(1+R/200)^2

$$\frac { 3600 } { 2500 } = ( \frac { 1 + R } { 200 } ) ^ { 2 }$$

Solve for R

$$R=239$$

$$R=-241$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{3600}{2500} to lowest terms by extracting and canceling out 100.

$$\frac{36}{25}=\left(\frac{1+R}{200}\right)^{2}$$

To raise \frac{1+R}{200} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{36}{25}=\frac{\left(1+R\right)^{2}}{200^{2}}$$

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

$$\frac{36}{25}=\frac{1+2R+R^{2}}{200^{2}}$$

Calculate 200 to the power of 2 and get 40000.

$$\frac{36}{25}=\frac{1+2R+R^{2}}{40000}$$

Divide each term of 1+2R+R^{2} by 40000 to get \frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}.

$$\frac{36}{25}=\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}$$

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

$$\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}=\frac{36}{25}$$

Subtract \frac{36}{25} from both sides.

$$\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}-\frac{36}{25}=0$$

Subtract \frac{36}{25} from \frac{1}{40000} to get -\frac{57599}{40000}.

$$-\frac{57599}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}=0$$

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.

$$\frac{1}{40000}R^{2}+\frac{1}{20000}R-\frac{57599}{40000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{40000} for a, \frac{1}{20000} for b, and -\frac{57599}{40000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$R=\frac{-\frac{1}{20000}±\sqrt{\left(\frac{1}{20000}\right)^{2}-4\times \frac{1}{40000}\left(-\frac{57599}{40000}\right)}}{2\times \frac{1}{40000}}$$

Square \frac{1}{20000} by squaring both the numerator and the denominator of the fraction.

$$R=\frac{-\frac{1}{20000}±\sqrt{\frac{1}{400000000}-4\times \frac{1}{40000}\left(-\frac{57599}{40000}\right)}}{2\times \frac{1}{40000}}$$

Multiply -4 times \frac{1}{40000}.

$$R=\frac{-\frac{1}{20000}±\sqrt{\frac{1}{400000000}-\frac{1}{10000}\left(-\frac{57599}{40000}\right)}}{2\times \frac{1}{40000}}$$

Multiply -\frac{1}{10000} times -\frac{57599}{40000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$R=\frac{-\frac{1}{20000}±\sqrt{\frac{1+57599}{400000000}}}{2\times \frac{1}{40000}}$$

Add \frac{1}{400000000} to \frac{57599}{400000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$R=\frac{-\frac{1}{20000}±\sqrt{\frac{9}{62500}}}{2\times \frac{1}{40000}}$$

Take the square root of \frac{9}{62500}.

$$R=\frac{-\frac{1}{20000}±\frac{3}{250}}{2\times \frac{1}{40000}}$$

Multiply 2 times \frac{1}{40000}.

$$R=\frac{-\frac{1}{20000}±\frac{3}{250}}{\frac{1}{20000}}$$

Now solve the equation R=\frac{-\frac{1}{20000}±\frac{3}{250}}{\frac{1}{20000}} when ± is plus. Add -\frac{1}{20000} to \frac{3}{250} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$R=\frac{\frac{239}{20000}}{\frac{1}{20000}}$$

Divide \frac{239}{20000} by \frac{1}{20000} by multiplying \frac{239}{20000} by the reciprocal of \frac{1}{20000}.

$$R=239$$

Now solve the equation R=\frac{-\frac{1}{20000}±\frac{3}{250}}{\frac{1}{20000}} when ± is minus. Subtract \frac{3}{250} from -\frac{1}{20000} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$R=-\frac{\frac{241}{20000}}{\frac{1}{20000}}$$

Divide -\frac{241}{20000} by \frac{1}{20000} by multiplying -\frac{241}{20000} by the reciprocal of \frac{1}{20000}.

$$R=-241$$

The equation is now solved.

$$R=239 R=-241$$

Steps for Completing the Square

Reduce the fraction \frac{3600}{2500} to lowest terms by extracting and canceling out 100.

$$\frac{36}{25}=\left(\frac{1+R}{200}\right)^{2}$$

To raise \frac{1+R}{200} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{36}{25}=\frac{\left(1+R\right)^{2}}{200^{2}}$$

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

$$\frac{36}{25}=\frac{1+2R+R^{2}}{200^{2}}$$

Calculate 200 to the power of 2 and get 40000.

$$\frac{36}{25}=\frac{1+2R+R^{2}}{40000}$$

Divide each term of 1+2R+R^{2} by 40000 to get \frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}.

$$\frac{36}{25}=\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}$$

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

$$\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}=\frac{36}{25}$$

Subtract \frac{1}{40000} from both sides.

$$\frac{1}{20000}R+\frac{1}{40000}R^{2}=\frac{36}{25}-\frac{1}{40000}$$

Subtract \frac{1}{40000} from \frac{36}{25} to get \frac{57599}{40000}.

$$\frac{1}{20000}R+\frac{1}{40000}R^{2}=\frac{57599}{40000}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\frac{1}{40000}R^{2}+\frac{1}{20000}R=\frac{57599}{40000}$$

Multiply both sides by 40000.

$$\frac{\frac{1}{40000}R^{2}+\frac{1}{20000}R}{\frac{1}{40000}}=\frac{\frac{57599}{40000}}{\frac{1}{40000}}$$

Dividing by \frac{1}{40000} undoes the multiplication by \frac{1}{40000}.

$$R^{2}+\frac{\frac{1}{20000}}{\frac{1}{40000}}R=\frac{\frac{57599}{40000}}{\frac{1}{40000}}$$

Divide \frac{1}{20000} by \frac{1}{40000} by multiplying \frac{1}{20000} by the reciprocal of \frac{1}{40000}.

$$R^{2}+2R=\frac{\frac{57599}{40000}}{\frac{1}{40000}}$$

Divide \frac{57599}{40000} by \frac{1}{40000} by multiplying \frac{57599}{40000} by the reciprocal of \frac{1}{40000}.

$$R^{2}+2R=57599$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$R^{2}+2R+1^{2}=57599+1^{2}$$

Square 1.

$$R^{2}+2R+1=57599+1$$

Add 57599 to 1.

$$R^{2}+2R+1=57600$$

Factor R^{2}+2R+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(R+1\right)^{2}=57600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(R+1\right)^{2}}=\sqrt{57600}$$

Simplify.

$$R+1=240 R+1=-240$$

Subtract 1 from both sides of the equation.

$$R=239 R=-241$$