Solve (100/x-3)(x-2/120)div2x/120=2/3

$$( \frac { 100 } { x } - 3 ) ( \frac { x - 2 } { 120 } ) \div \frac { 2 x } { 120 } = \frac { 2 } { 3 }$$

Solve for x

$$x = \frac{\sqrt{17481} + 159}{13} \approx 22.401210215$$

$$x = \frac{159 - \sqrt{17481}}{13} \approx 2.060328246$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x}{x}.

$$\frac{\left(\frac{100}{x}-\frac{3x}{x}\right)\times \frac{x-2}{120}}{\frac{2x}{120}}=\frac{2}{3}$$

Since \frac{100}{x} and \frac{3x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{100-3x}{x}\times \frac{x-2}{120}}{\frac{2x}{120}}=\frac{2}{3}$$

Multiply \frac{100-3x}{x} times \frac{x-2}{120} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(100-3x\right)\left(x-2\right)}{x\times 120}}{\frac{2x}{120}}=\frac{2}{3}$$

Divide 2x by 120 to get \frac{1}{60}x.

$$\frac{\frac{\left(100-3x\right)\left(x-2\right)}{x\times 120}}{\frac{1}{60}x}=\frac{2}{3}$$

Express \frac{\frac{\left(100-3x\right)\left(x-2\right)}{x\times 120}}{\frac{1}{60}x} as a single fraction.

$$\frac{\left(100-3x\right)\left(x-2\right)}{x\times 120\times \frac{1}{60}x}=\frac{2}{3}$$

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

$$\frac{\left(100-3x\right)\left(x-2\right)}{x^{2}\times 120\times \frac{1}{60}}=\frac{2}{3}$$

Multiply 120 and \frac{1}{60} to get 2.

$$\frac{\left(100-3x\right)\left(x-2\right)}{x^{2}\times 2}=\frac{2}{3}$$

Use the distributive property to multiply 100-3x by x-2 and combine like terms.

$$\frac{106x-200-3x^{2}}{x^{2}\times 2}=\frac{2}{3}$$

Subtract \frac{2}{3} from both sides.

$$\frac{106x-200-3x^{2}}{x^{2}\times 2}-\frac{2}{3}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}\times 2 and 3 is 6x^{2}. Multiply \frac{106x-200-3x^{2}}{x^{2}\times 2} times \frac{3}{3}. Multiply \frac{2}{3} times \frac{2x^{2}}{2x^{2}}.

$$\frac{3\left(106x-200-3x^{2}\right)}{6x^{2}}-\frac{2\times 2x^{2}}{6x^{2}}=0$$

Since \frac{3\left(106x-200-3x^{2}\right)}{6x^{2}} and \frac{2\times 2x^{2}}{6x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(106x-200-3x^{2}\right)-2\times 2x^{2}}{6x^{2}}=0$$

Do the multiplications in 3\left(106x-200-3x^{2}\right)-2\times 2x^{2}.

$$\frac{318x-600-9x^{2}-4x^{2}}{6x^{2}}=0$$

Combine like terms in 318x-600-9x^{2}-4x^{2}.

$$\frac{318x-600-13x^{2}}{6x^{2}}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x^{2}.

$$318x-600-13x^{2}=0$$

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.

$$-13x^{2}+318x-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -13 for a, 318 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-318±\sqrt{318^{2}-4\left(-13\right)\left(-600\right)}}{2\left(-13\right)}$$

Square 318.

$$x=\frac{-318±\sqrt{101124-4\left(-13\right)\left(-600\right)}}{2\left(-13\right)}$$

Multiply -4 times -13.

$$x=\frac{-318±\sqrt{101124+52\left(-600\right)}}{2\left(-13\right)}$$

Multiply 52 times -600.

$$x=\frac{-318±\sqrt{101124-31200}}{2\left(-13\right)}$$

Add 101124 to -31200.

$$x=\frac{-318±\sqrt{69924}}{2\left(-13\right)}$$

Take the square root of 69924.

$$x=\frac{-318±2\sqrt{17481}}{2\left(-13\right)}$$

Multiply 2 times -13.

$$x=\frac{-318±2\sqrt{17481}}{-26}$$

Now solve the equation x=\frac{-318±2\sqrt{17481}}{-26} when ± is plus. Add -318 to 2\sqrt{17481}.

$$x=\frac{2\sqrt{17481}-318}{-26}$$

Divide -318+2\sqrt{17481} by -26.

$$x=\frac{159-\sqrt{17481}}{13}$$

Now solve the equation x=\frac{-318±2\sqrt{17481}}{-26} when ± is minus. Subtract 2\sqrt{17481} from -318.

$$x=\frac{-2\sqrt{17481}-318}{-26}$$

Divide -318-2\sqrt{17481} by -26.

$$x=\frac{\sqrt{17481}+159}{13}$$

The equation is now solved.

$$x=\frac{159-\sqrt{17481}}{13} x=\frac{\sqrt{17481}+159}{13}$$

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x}{x}.

$$\frac{\left(\frac{100}{x}-\frac{3x}{x}\right)\times \frac{x-2}{120}}{\frac{2x}{120}}=\frac{2}{3}$$

Since \frac{100}{x} and \frac{3x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{100-3x}{x}\times \frac{x-2}{120}}{\frac{2x}{120}}=\frac{2}{3}$$

Multiply \frac{100-3x}{x} times \frac{x-2}{120} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(100-3x\right)\left(x-2\right)}{x\times 120}}{\frac{2x}{120}}=\frac{2}{3}$$

Divide 2x by 120 to get \frac{1}{60}x.

$$\frac{\frac{\left(100-3x\right)\left(x-2\right)}{x\times 120}}{\frac{1}{60}x}=\frac{2}{3}$$

Express \frac{\frac{\left(100-3x\right)\left(x-2\right)}{x\times 120}}{\frac{1}{60}x} as a single fraction.

$$\frac{\left(100-3x\right)\left(x-2\right)}{x\times 120\times \frac{1}{60}x}=\frac{2}{3}$$

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

$$\frac{\left(100-3x\right)\left(x-2\right)}{x^{2}\times 120\times \frac{1}{60}}=\frac{2}{3}$$

Multiply 120 and \frac{1}{60} to get 2.

$$\frac{\left(100-3x\right)\left(x-2\right)}{x^{2}\times 2}=\frac{2}{3}$$

Use the distributive property to multiply 100-3x by x-2 and combine like terms.

$$\frac{106x-200-3x^{2}}{x^{2}\times 2}=\frac{2}{3}$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x^{2}, the least common multiple of x^{2}\times 2,3.

$$3\left(106x-200-3x^{2}\right)=4x^{2}$$

Use the distributive property to multiply 3 by 106x-200-3x^{2}.

$$318x-600-9x^{2}=4x^{2}$$

Subtract 4x^{2} from both sides.

$$318x-600-9x^{2}-4x^{2}=0$$

Combine -9x^{2} and -4x^{2} to get -13x^{2}.

$$318x-600-13x^{2}=0$$

Add 600 to both sides. Anything plus zero gives itself.

$$318x-13x^{2}=600$$

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.

$$-13x^{2}+318x=600$$

Divide both sides by -13.

$$\frac{-13x^{2}+318x}{-13}=\frac{600}{-13}$$

Dividing by -13 undoes the multiplication by -13.

$$x^{2}+\frac{318}{-13}x=\frac{600}{-13}$$

Divide 318 by -13.

$$x^{2}-\frac{318}{13}x=\frac{600}{-13}$$

Divide 600 by -13.

$$x^{2}-\frac{318}{13}x=-\frac{600}{13}$$

Divide -\frac{318}{13}, the coefficient of the x term, by 2 to get -\frac{159}{13}. Then add the square of -\frac{159}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{318}{13}x+\left(-\frac{159}{13}\right)^{2}=-\frac{600}{13}+\left(-\frac{159}{13}\right)^{2}$$

Square -\frac{159}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{318}{13}x+\frac{25281}{169}=-\frac{600}{13}+\frac{25281}{169}$$

Add -\frac{600}{13} to \frac{25281}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{318}{13}x+\frac{25281}{169}=\frac{17481}{169}$$

Factor x^{2}-\frac{318}{13}x+\frac{25281}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{159}{13}\right)^{2}=\frac{17481}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{159}{13}\right)^{2}}=\sqrt{\frac{17481}{169}}$$

Simplify.

$$x-\frac{159}{13}=\frac{\sqrt{17481}}{13} x-\frac{159}{13}=-\frac{\sqrt{17481}}{13}$$

Add \frac{159}{13} to both sides of the equation.

$$x=\frac{\sqrt{17481}+159}{13} x=\frac{159-\sqrt{17481}}{13}$$