Solve 20+30r/frac{1{30}+1/r}=40

$$20 + \frac { 30 r } { \frac { 1 } { 30 } + \frac { 1 } { r } } = 40$$

Solve for r

$$r=\frac{\sqrt{5401}+1}{90}\approx 0.82768329$$

$$r=\frac{1-\sqrt{5401}}{90}\approx -0.805461068$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 30 and r is 30r. Multiply \frac{1}{30} times \frac{r}{r}. Multiply \frac{1}{r} times \frac{30}{30}.

$$20+\frac{30r}{\frac{r}{30r}+\frac{30}{30r}}=40$$

Since \frac{r}{30r} and \frac{30}{30r} have the same denominator, add them by adding their numerators.

$$20+\frac{30r}{\frac{r+30}{30r}}=40$$

Variable r cannot be equal to 0 since division by zero is not defined. Divide 30r by \frac{r+30}{30r} by multiplying 30r by the reciprocal of \frac{r+30}{30r}.

$$20+\frac{30r\times 30r}{r+30}=40$$

Multiply r and r to get r^{2}.

$$20+\frac{30r^{2}\times 30}{r+30}=40$$

Multiply 30 and 30 to get 900.

$$20+\frac{900r^{2}}{r+30}=40$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 20 times \frac{r+30}{r+30}.

$$\frac{20\left(r+30\right)}{r+30}+\frac{900r^{2}}{r+30}=40$$

Since \frac{20\left(r+30\right)}{r+30} and \frac{900r^{2}}{r+30} have the same denominator, add them by adding their numerators.

$$\frac{20\left(r+30\right)+900r^{2}}{r+30}=40$$

Do the multiplications in 20\left(r+30\right)+900r^{2}.

$$\frac{20r+600+900r^{2}}{r+30}=40$$

Subtract 40 from both sides.

$$\frac{20r+600+900r^{2}}{r+30}-40=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 40 times \frac{r+30}{r+30}.

$$\frac{20r+600+900r^{2}}{r+30}-\frac{40\left(r+30\right)}{r+30}=0$$

Since \frac{20r+600+900r^{2}}{r+30} and \frac{40\left(r+30\right)}{r+30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{20r+600+900r^{2}-40\left(r+30\right)}{r+30}=0$$

Do the multiplications in 20r+600+900r^{2}-40\left(r+30\right).

$$\frac{20r+600+900r^{2}-40r-1200}{r+30}=0$$

Combine like terms in 20r+600+900r^{2}-40r-1200.

$$\frac{-20r-600+900r^{2}}{r+30}=0$$

Variable r cannot be equal to -30 since division by zero is not defined. Multiply both sides of the equation by r+30.

$$-20r-600+900r^{2}=0$$

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.

$$900r^{2}-20r-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 900 for a, -20 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 900\left(-600\right)}}{2\times 900}$$

Square -20.

$$r=\frac{-\left(-20\right)±\sqrt{400-4\times 900\left(-600\right)}}{2\times 900}$$

Multiply -4 times 900.

$$r=\frac{-\left(-20\right)±\sqrt{400-3600\left(-600\right)}}{2\times 900}$$

Multiply -3600 times -600.

$$r=\frac{-\left(-20\right)±\sqrt{400+2160000}}{2\times 900}$$

Add 400 to 2160000.

$$r=\frac{-\left(-20\right)±\sqrt{2160400}}{2\times 900}$$

Take the square root of 2160400.

$$r=\frac{-\left(-20\right)±20\sqrt{5401}}{2\times 900}$$

The opposite of -20 is 20.

$$r=\frac{20±20\sqrt{5401}}{2\times 900}$$

Multiply 2 times 900.

$$r=\frac{20±20\sqrt{5401}}{1800}$$

Now solve the equation r=\frac{20±20\sqrt{5401}}{1800} when ± is plus. Add 20 to 20\sqrt{5401}.

$$r=\frac{20\sqrt{5401}+20}{1800}$$

Divide 20+20\sqrt{5401} by 1800.

$$r=\frac{\sqrt{5401}+1}{90}$$

Now solve the equation r=\frac{20±20\sqrt{5401}}{1800} when ± is minus. Subtract 20\sqrt{5401} from 20.

$$r=\frac{20-20\sqrt{5401}}{1800}$$

Divide 20-20\sqrt{5401} by 1800.

$$r=\frac{1-\sqrt{5401}}{90}$$

The equation is now solved.

$$r=\frac{\sqrt{5401}+1}{90} r=\frac{1-\sqrt{5401}}{90}$$

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 30 and r is 30r. Multiply \frac{1}{30} times \frac{r}{r}. Multiply \frac{1}{r} times \frac{30}{30}.

$$20+\frac{30r}{\frac{r}{30r}+\frac{30}{30r}}=40$$

Since \frac{r}{30r} and \frac{30}{30r} have the same denominator, add them by adding their numerators.

$$20+\frac{30r}{\frac{r+30}{30r}}=40$$

Variable r cannot be equal to 0 since division by zero is not defined. Divide 30r by \frac{r+30}{30r} by multiplying 30r by the reciprocal of \frac{r+30}{30r}.

$$20+\frac{30r\times 30r}{r+30}=40$$

Multiply r and r to get r^{2}.

$$20+\frac{30r^{2}\times 30}{r+30}=40$$

Multiply 30 and 30 to get 900.

$$20+\frac{900r^{2}}{r+30}=40$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 20 times \frac{r+30}{r+30}.

$$\frac{20\left(r+30\right)}{r+30}+\frac{900r^{2}}{r+30}=40$$

Since \frac{20\left(r+30\right)}{r+30} and \frac{900r^{2}}{r+30} have the same denominator, add them by adding their numerators.

$$\frac{20\left(r+30\right)+900r^{2}}{r+30}=40$$

Do the multiplications in 20\left(r+30\right)+900r^{2}.

$$\frac{20r+600+900r^{2}}{r+30}=40$$

Variable r cannot be equal to -30 since division by zero is not defined. Multiply both sides of the equation by r+30.

$$20r+600+900r^{2}=40\left(r+30\right)$$

Use the distributive property to multiply 40 by r+30.

$$20r+600+900r^{2}=40r+1200$$

Subtract 40r from both sides.

$$20r+600+900r^{2}-40r=1200$$

Combine 20r and -40r to get -20r.

$$-20r+600+900r^{2}=1200$$

Subtract 600 from both sides.

$$-20r+900r^{2}=1200-600$$

Subtract 600 from 1200 to get 600.

$$-20r+900r^{2}=600$$

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.

$$900r^{2}-20r=600$$

Divide both sides by 900.

$$\frac{900r^{2}-20r}{900}=\frac{600}{900}$$

Dividing by 900 undoes the multiplication by 900.

$$r^{2}+\left(-\frac{20}{900}\right)r=\frac{600}{900}$$

Reduce the fraction \frac{-20}{900} to lowest terms by extracting and canceling out 20.

$$r^{2}-\frac{1}{45}r=\frac{600}{900}$$

Reduce the fraction \frac{600}{900} to lowest terms by extracting and canceling out 300.

$$r^{2}-\frac{1}{45}r=\frac{2}{3}$$

Divide -\frac{1}{45}, the coefficient of the x term, by 2 to get -\frac{1}{90}. Then add the square of -\frac{1}{90} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}-\frac{1}{45}r+\left(-\frac{1}{90}\right)^{2}=\frac{2}{3}+\left(-\frac{1}{90}\right)^{2}$$

Square -\frac{1}{90} by squaring both the numerator and the denominator of the fraction.

$$r^{2}-\frac{1}{45}r+\frac{1}{8100}=\frac{2}{3}+\frac{1}{8100}$$

Add \frac{2}{3} to \frac{1}{8100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$r^{2}-\frac{1}{45}r+\frac{1}{8100}=\frac{5401}{8100}$$

Factor r^{2}-\frac{1}{45}r+\frac{1}{8100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r-\frac{1}{90}\right)^{2}=\frac{5401}{8100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-\frac{1}{90}\right)^{2}}=\sqrt{\frac{5401}{8100}}$$

Simplify.

$$r-\frac{1}{90}=\frac{\sqrt{5401}}{90} r-\frac{1}{90}=-\frac{\sqrt{5401}}{90}$$

Add \frac{1}{90} to both sides of the equation.

$$r=\frac{\sqrt{5401}+1}{90} r=\frac{1-\sqrt{5401}}{90}$$