Solve 100=0.5/1+frac{r{2}}+0.5/{left(1+frac{r{2}right)}^2}+110/{left(1+frac{r{2}right)}^2}

$$100 = \frac{ 0.5 }{ 1+ \frac{ r }{ 2 } } + \frac{ 0.5 }{ { \left(1+ \frac{ r }{ 2 } \right) }^{ 2 } } + \frac{ 110 }{ { \left(1+ \frac{ r }{ 2 } \right) }^{ 2 } }$$

Solve for r

$$r=\frac{-\sqrt{176801}-399}{200}\approx -4.09738555$$

$$r=\frac{\sqrt{176801}-399}{200}\approx 0.10738555$$

Steps Using the Quadratic Formula

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

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{1+2\times \frac{r}{2}+\left(\frac{r}{2}\right)^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Express 2\times \frac{r}{2} as a single fraction.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{1+\frac{2r}{2}+\left(\frac{r}{2}\right)^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Cancel out 2 and 2.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{1+r+\left(\frac{r}{2}\right)^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

To raise \frac{r}{2} to a power, raise both numerator and denominator to the power and then divide.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{1+r+\frac{r^{2}}{2^{2}}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

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

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{\frac{\left(1+r\right)\times 2^{2}}{2^{2}}+\frac{r^{2}}{2^{2}}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Since \frac{\left(1+r\right)\times 2^{2}}{2^{2}} and \frac{r^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{\frac{\left(1+r\right)\times 2^{2}+r^{2}}{2^{2}}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Do the multiplications in \left(1+r\right)\times 2^{2}+r^{2}.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{\frac{4+4r+r^{2}}{2^{2}}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Divide 0.5 by \frac{4+4r+r^{2}}{2^{2}} by multiplying 0.5 by the reciprocal of \frac{4+4r+r^{2}}{2^{2}}.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5\times 2^{2}}{4+4r+r^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5\times 4}{4+4r+r^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Multiply 0.5 and 4 to get 2.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

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

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{1+2\times \frac{r}{2}+\left(\frac{r}{2}\right)^{2}}$$

Express 2\times \frac{r}{2} as a single fraction.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{1+\frac{2r}{2}+\left(\frac{r}{2}\right)^{2}}$$

Cancel out 2 and 2.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{1+r+\left(\frac{r}{2}\right)^{2}}$$

To raise \frac{r}{2} to a power, raise both numerator and denominator to the power and then divide.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{1+r+\frac{r^{2}}{2^{2}}}$$

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

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{\frac{\left(1+r\right)\times 2^{2}}{2^{2}}+\frac{r^{2}}{2^{2}}}$$

Since \frac{\left(1+r\right)\times 2^{2}}{2^{2}} and \frac{r^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{\frac{\left(1+r\right)\times 2^{2}+r^{2}}{2^{2}}}$$

Do the multiplications in \left(1+r\right)\times 2^{2}+r^{2}.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{\frac{4+4r+r^{2}}{2^{2}}}$$

Divide 110 by \frac{4+4r+r^{2}}{2^{2}} by multiplying 110 by the reciprocal of \frac{4+4r+r^{2}}{2^{2}}.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110\times 2^{2}}{4+4r+r^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110\times 4}{4+4r+r^{2}}$$

Multiply 110 and 4 to get 440.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{440}{4+4r+r^{2}}$$

Since \frac{2}{4+4r+r^{2}} and \frac{440}{4+4r+r^{2}} have the same denominator, add them by adding their numerators. Add 2 and 440 to get 442.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{442}{4+4r+r^{2}}$$

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

$$\frac{0.5}{1+\frac{r}{2}}+\frac{442}{4+4r+r^{2}}=100$$

Subtract 100 from both sides.

$$\frac{0.5}{1+\frac{r}{2}}+\frac{442}{4+4r+r^{2}}-100=0$$

Factor 4+4r+r^{2}.

$$\frac{0.5}{1+\frac{r}{2}}+\frac{442}{\left(r+2\right)^{2}}-100=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 100 times \frac{\left(r+2\right)^{2}}{\left(r+2\right)^{2}}.

$$\frac{0.5}{1+\frac{r}{2}}+\frac{442}{\left(r+2\right)^{2}}-\frac{100\left(r+2\right)^{2}}{\left(r+2\right)^{2}}=0$$

Since \frac{442}{\left(r+2\right)^{2}} and \frac{100\left(r+2\right)^{2}}{\left(r+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{0.5}{1+\frac{r}{2}}+\frac{442-100\left(r+2\right)^{2}}{\left(r+2\right)^{2}}=0$$

Do the multiplications in 442-100\left(r+2\right)^{2}.

$$\frac{0.5}{1+\frac{r}{2}}+\frac{442-100r^{2}-400r-400}{\left(r+2\right)^{2}}=0$$

Combine like terms in 442-100r^{2}-400r-400.

$$\frac{0.5}{1+\frac{r}{2}}+\frac{42-100r^{2}-400r}{\left(r+2\right)^{2}}=0$$

Variable r cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by \left(r+2\right)^{2}.

$$\left(2r+4\right)\times 0.5+42-100r^{2}-400r=0$$

Reorder the terms.

$$-100r^{2}+0.5\left(2r+4\right)-400r+42=0$$

Use the distributive property to multiply 0.5 by 2r+4.

$$-100r^{2}+r+2-400r+42=0$$

Combine r and -400r to get -399r.

$$-100r^{2}-399r+2+42=0$$

Add 2 and 42 to get 44.

$$-100r^{2}-399r+44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, -399 for b, and 44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-399\right)±\sqrt{\left(-399\right)^{2}-4\left(-100\right)\times 44}}{2\left(-100\right)}$$

Square -399.

$$r=\frac{-\left(-399\right)±\sqrt{159201-4\left(-100\right)\times 44}}{2\left(-100\right)}$$

Multiply -4 times -100.

$$r=\frac{-\left(-399\right)±\sqrt{159201+400\times 44}}{2\left(-100\right)}$$

Multiply 400 times 44.

$$r=\frac{-\left(-399\right)±\sqrt{159201+17600}}{2\left(-100\right)}$$

Add 159201 to 17600.

$$r=\frac{-\left(-399\right)±\sqrt{176801}}{2\left(-100\right)}$$

The opposite of -399 is 399.

$$r=\frac{399±\sqrt{176801}}{2\left(-100\right)}$$

Multiply 2 times -100.

$$r=\frac{399±\sqrt{176801}}{-200}$$

Now solve the equation r=\frac{399±\sqrt{176801}}{-200} when ± is plus. Add 399 to \sqrt{176801}.

$$r=\frac{\sqrt{176801}+399}{-200}$$

Divide 399+\sqrt{176801} by -200.

$$r=\frac{-\sqrt{176801}-399}{200}$$

Now solve the equation r=\frac{399±\sqrt{176801}}{-200} when ± is minus. Subtract \sqrt{176801} from 399.

$$r=\frac{399-\sqrt{176801}}{-200}$$

Divide 399-\sqrt{176801} by -200.

$$r=\frac{\sqrt{176801}-399}{200}$$

The equation is now solved.

$$r=\frac{-\sqrt{176801}-399}{200} r=\frac{\sqrt{176801}-399}{200}$$

Steps for Completing the Square

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

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{1+2\times \frac{r}{2}+\left(\frac{r}{2}\right)^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Express 2\times \frac{r}{2} as a single fraction.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{1+\frac{2r}{2}+\left(\frac{r}{2}\right)^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Cancel out 2 and 2.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{1+r+\left(\frac{r}{2}\right)^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

To raise \frac{r}{2} to a power, raise both numerator and denominator to the power and then divide.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{1+r+\frac{r^{2}}{2^{2}}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

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

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{\frac{\left(1+r\right)\times 2^{2}}{2^{2}}+\frac{r^{2}}{2^{2}}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Since \frac{\left(1+r\right)\times 2^{2}}{2^{2}} and \frac{r^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{\frac{\left(1+r\right)\times 2^{2}+r^{2}}{2^{2}}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Do the multiplications in \left(1+r\right)\times 2^{2}+r^{2}.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5}{\frac{4+4r+r^{2}}{2^{2}}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Divide 0.5 by \frac{4+4r+r^{2}}{2^{2}} by multiplying 0.5 by the reciprocal of \frac{4+4r+r^{2}}{2^{2}}.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5\times 2^{2}}{4+4r+r^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{0.5\times 4}{4+4r+r^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

Multiply 0.5 and 4 to get 2.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{\left(1+\frac{r}{2}\right)^{2}}$$

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

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{1+2\times \frac{r}{2}+\left(\frac{r}{2}\right)^{2}}$$

Express 2\times \frac{r}{2} as a single fraction.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{1+\frac{2r}{2}+\left(\frac{r}{2}\right)^{2}}$$

Cancel out 2 and 2.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{1+r+\left(\frac{r}{2}\right)^{2}}$$

To raise \frac{r}{2} to a power, raise both numerator and denominator to the power and then divide.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{1+r+\frac{r^{2}}{2^{2}}}$$

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

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{\frac{\left(1+r\right)\times 2^{2}}{2^{2}}+\frac{r^{2}}{2^{2}}}$$

Since \frac{\left(1+r\right)\times 2^{2}}{2^{2}} and \frac{r^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{\frac{\left(1+r\right)\times 2^{2}+r^{2}}{2^{2}}}$$

Do the multiplications in \left(1+r\right)\times 2^{2}+r^{2}.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110}{\frac{4+4r+r^{2}}{2^{2}}}$$

Divide 110 by \frac{4+4r+r^{2}}{2^{2}} by multiplying 110 by the reciprocal of \frac{4+4r+r^{2}}{2^{2}}.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110\times 2^{2}}{4+4r+r^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{110\times 4}{4+4r+r^{2}}$$

Multiply 110 and 4 to get 440.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{2}{4+4r+r^{2}}+\frac{440}{4+4r+r^{2}}$$

Since \frac{2}{4+4r+r^{2}} and \frac{440}{4+4r+r^{2}} have the same denominator, add them by adding their numerators. Add 2 and 440 to get 442.

$$100=\frac{0.5}{1+\frac{r}{2}}+\frac{442}{4+4r+r^{2}}$$

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

$$\frac{0.5}{1+\frac{r}{2}}+\frac{442}{4+4r+r^{2}}=100$$

Variable r cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by \left(r+2\right)^{2}.

$$\left(2r+4\right)\times 0.5+442=100\left(r+2\right)^{2}$$

Use the distributive property to multiply 2r+4 by 0.5.

$$r+2+442=100\left(r+2\right)^{2}$$

Add 2 and 442 to get 444.

$$r+444=100\left(r+2\right)^{2}$$

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

$$r+444=100\left(r^{2}+4r+4\right)$$

Use the distributive property to multiply 100 by r^{2}+4r+4.

$$r+444=100r^{2}+400r+400$$

Subtract 100r^{2} from both sides.

$$r+444-100r^{2}=400r+400$$

Subtract 400r from both sides.

$$r+444-100r^{2}-400r=400$$

Combine r and -400r to get -399r.

$$-399r+444-100r^{2}=400$$

Subtract 444 from both sides.

$$-399r-100r^{2}=400-444$$

Subtract 444 from 400 to get -44.

$$-399r-100r^{2}=-44$$

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.

$$-100r^{2}-399r=-44$$

Divide both sides by -100.

$$\frac{-100r^{2}-399r}{-100}=-\frac{44}{-100}$$

Dividing by -100 undoes the multiplication by -100.

$$r^{2}+\left(-\frac{399}{-100}\right)r=-\frac{44}{-100}$$

Divide -399 by -100.

$$r^{2}+\frac{399}{100}r=-\frac{44}{-100}$$

Reduce the fraction \frac{-44}{-100} to lowest terms by extracting and canceling out 4.

$$r^{2}+\frac{399}{100}r=\frac{11}{25}$$

Divide \frac{399}{100}, the coefficient of the x term, by 2 to get \frac{399}{200}. Then add the square of \frac{399}{200} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+\frac{399}{100}r+\left(\frac{399}{200}\right)^{2}=\frac{11}{25}+\left(\frac{399}{200}\right)^{2}$$

Square \frac{399}{200} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+\frac{399}{100}r+\frac{159201}{40000}=\frac{11}{25}+\frac{159201}{40000}$$

Add \frac{11}{25} to \frac{159201}{40000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$r^{2}+\frac{399}{100}r+\frac{159201}{40000}=\frac{176801}{40000}$$

Factor r^{2}+\frac{399}{100}r+\frac{159201}{40000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+\frac{399}{200}\right)^{2}=\frac{176801}{40000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{399}{200}\right)^{2}}=\sqrt{\frac{176801}{40000}}$$

Simplify.

$$r+\frac{399}{200}=\frac{\sqrt{176801}}{200} r+\frac{399}{200}=-\frac{\sqrt{176801}}{200}$$

Subtract \frac{399}{200} from both sides of the equation.

$$r=\frac{\sqrt{176801}-399}{200} r=\frac{-\sqrt{176801}-399}{200}$$