Solve 1-b/frac{b{5}}+b/frac{6{5}-72/5(12+b)}=60

$$\frac { 1 - b } { \frac { b } { 5 } } + \frac { b } { \frac { 6 } { 5 } - \frac { 72 } { 5 ( 12 + b ) } } = 60$$

Solve for b

$$b=19\sqrt{3}+33\approx 65.908965344$$

$$b=33-19\sqrt{3}\approx 0.091034656$$

Steps Using the Quadratic Formula

Divide 1-b by \frac{b}{5} by multiplying 1-b by the reciprocal of \frac{b}{5}.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6}{5}-\frac{72}{5\left(12+b\right)}}=60$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 5\left(12+b\right) is 5\left(b+12\right). Multiply \frac{6}{5} times \frac{b+12}{b+12}.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6\left(b+12\right)}{5\left(b+12\right)}-\frac{72}{5\left(b+12\right)}}=60$$

Since \frac{6\left(b+12\right)}{5\left(b+12\right)} and \frac{72}{5\left(b+12\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6\left(b+12\right)-72}{5\left(b+12\right)}}=60$$

Do the multiplications in 6\left(b+12\right)-72.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6b+72-72}{5\left(b+12\right)}}=60$$

Combine like terms in 6b+72-72.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6b}{5\left(b+12\right)}}=60$$

Variable b cannot be equal to -12 since division by zero is not defined. Divide b by \frac{6b}{5\left(b+12\right)} by multiplying b by the reciprocal of \frac{6b}{5\left(b+12\right)}.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b\times 5\left(b+12\right)}{6b}=60$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and 6b is 6b. Multiply \frac{\left(1-b\right)\times 5}{b} times \frac{6}{6}.

$$\frac{6\left(1-b\right)\times 5}{6b}+\frac{b\times 5\left(b+12\right)}{6b}=60$$

Since \frac{6\left(1-b\right)\times 5}{6b} and \frac{b\times 5\left(b+12\right)}{6b} have the same denominator, add them by adding their numerators.

$$\frac{6\left(1-b\right)\times 5+b\times 5\left(b+12\right)}{6b}=60$$

Do the multiplications in 6\left(1-b\right)\times 5+b\times 5\left(b+12\right).

$$\frac{30-30b+5b^{2}+60b}{6b}=60$$

Combine like terms in 30-30b+5b^{2}+60b.

$$\frac{30+30b+5b^{2}}{6b}=60$$

Subtract 60 from both sides.

$$\frac{30+30b+5b^{2}}{6b}-60=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 60 times \frac{6b}{6b}.

$$\frac{30+30b+5b^{2}}{6b}-\frac{60\times 6b}{6b}=0$$

Since \frac{30+30b+5b^{2}}{6b} and \frac{60\times 6b}{6b} have the same denominator, subtract them by subtracting their numerators.

$$\frac{30+30b+5b^{2}-60\times 6b}{6b}=0$$

Do the multiplications in 30+30b+5b^{2}-60\times 6b.

$$\frac{30+30b+5b^{2}-360b}{6b}=0$$

Combine like terms in 30+30b+5b^{2}-360b.

$$\frac{30-330b+5b^{2}}{6b}=0$$

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6b.

$$30-330b+5b^{2}=0$$

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.

$$5b^{2}-330b+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -330 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-330\right)±\sqrt{\left(-330\right)^{2}-4\times 5\times 30}}{2\times 5}$$

Square -330.

$$b=\frac{-\left(-330\right)±\sqrt{108900-4\times 5\times 30}}{2\times 5}$$

Multiply -4 times 5.

$$b=\frac{-\left(-330\right)±\sqrt{108900-20\times 30}}{2\times 5}$$

Multiply -20 times 30.

$$b=\frac{-\left(-330\right)±\sqrt{108900-600}}{2\times 5}$$

Add 108900 to -600.

$$b=\frac{-\left(-330\right)±\sqrt{108300}}{2\times 5}$$

Take the square root of 108300.

$$b=\frac{-\left(-330\right)±190\sqrt{3}}{2\times 5}$$

The opposite of -330 is 330.

$$b=\frac{330±190\sqrt{3}}{2\times 5}$$

Multiply 2 times 5.

$$b=\frac{330±190\sqrt{3}}{10}$$

Now solve the equation b=\frac{330±190\sqrt{3}}{10} when ± is plus. Add 330 to 190\sqrt{3}.

$$b=\frac{190\sqrt{3}+330}{10}$$

Divide 330+190\sqrt{3} by 10.

$$b=19\sqrt{3}+33$$

Now solve the equation b=\frac{330±190\sqrt{3}}{10} when ± is minus. Subtract 190\sqrt{3} from 330.

$$b=\frac{330-190\sqrt{3}}{10}$$

Divide 330-190\sqrt{3} by 10.

$$b=33-19\sqrt{3}$$

The equation is now solved.

$$b=19\sqrt{3}+33 b=33-19\sqrt{3}$$

Steps for Completing the Square

Divide 1-b by \frac{b}{5} by multiplying 1-b by the reciprocal of \frac{b}{5}.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6}{5}-\frac{72}{5\left(12+b\right)}}=60$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 5\left(12+b\right) is 5\left(b+12\right). Multiply \frac{6}{5} times \frac{b+12}{b+12}.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6\left(b+12\right)}{5\left(b+12\right)}-\frac{72}{5\left(b+12\right)}}=60$$

Since \frac{6\left(b+12\right)}{5\left(b+12\right)} and \frac{72}{5\left(b+12\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6\left(b+12\right)-72}{5\left(b+12\right)}}=60$$

Do the multiplications in 6\left(b+12\right)-72.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6b+72-72}{5\left(b+12\right)}}=60$$

Combine like terms in 6b+72-72.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b}{\frac{6b}{5\left(b+12\right)}}=60$$

Variable b cannot be equal to -12 since division by zero is not defined. Divide b by \frac{6b}{5\left(b+12\right)} by multiplying b by the reciprocal of \frac{6b}{5\left(b+12\right)}.

$$\frac{\left(1-b\right)\times 5}{b}+\frac{b\times 5\left(b+12\right)}{6b}=60$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and 6b is 6b. Multiply \frac{\left(1-b\right)\times 5}{b} times \frac{6}{6}.

$$\frac{6\left(1-b\right)\times 5}{6b}+\frac{b\times 5\left(b+12\right)}{6b}=60$$

Since \frac{6\left(1-b\right)\times 5}{6b} and \frac{b\times 5\left(b+12\right)}{6b} have the same denominator, add them by adding their numerators.

$$\frac{6\left(1-b\right)\times 5+b\times 5\left(b+12\right)}{6b}=60$$

Do the multiplications in 6\left(1-b\right)\times 5+b\times 5\left(b+12\right).

$$\frac{30-30b+5b^{2}+60b}{6b}=60$$

Combine like terms in 30-30b+5b^{2}+60b.

$$\frac{30+30b+5b^{2}}{6b}=60$$

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6b.

$$30+30b+5b^{2}=360b$$

Subtract 360b from both sides.

$$30+30b+5b^{2}-360b=0$$

Combine 30b and -360b to get -330b.

$$30-330b+5b^{2}=0$$

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

$$-330b+5b^{2}=-30$$

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.

$$5b^{2}-330b=-30$$

Divide both sides by 5.

$$\frac{5b^{2}-330b}{5}=-\frac{30}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$b^{2}+\left(-\frac{330}{5}\right)b=-\frac{30}{5}$$

Divide -330 by 5.

$$b^{2}-66b=-\frac{30}{5}$$

Divide -30 by 5.

$$b^{2}-66b=-6$$

Divide -66, the coefficient of the x term, by 2 to get -33. Then add the square of -33 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-66b+\left(-33\right)^{2}=-6+\left(-33\right)^{2}$$

Square -33.

$$b^{2}-66b+1089=-6+1089$$

Add -6 to 1089.

$$b^{2}-66b+1089=1083$$

Factor b^{2}-66b+1089. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-33\right)^{2}=1083$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-33\right)^{2}}=\sqrt{1083}$$

Simplify.

$$b-33=19\sqrt{3} b-33=-19\sqrt{3}$$

Add 33 to both sides of the equation.

$$b=19\sqrt{3}+33 b=33-19\sqrt{3}$$