Solve 30/x+20left(1/40+1/xright)=1

$$\frac{ 30 }{ x } +20 \left( \frac{ 1 }{ 40 } + \frac{ 1 }{ x } \right) = 1$$

Solve for x

$$x=100$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 40x, the least common multiple of x,40.

$$40\times 30+20\left(\frac{1}{40}+\frac{1}{x}\right)\times 40x=40x$$

Multiply 40 and 30 to get 1200.

$$1200+20\left(\frac{1}{40}+\frac{1}{x}\right)\times 40x=40x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 40 and x is 40x. Multiply \frac{1}{40} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{40}{40}.

$$1200+20\left(\frac{x}{40x}+\frac{40}{40x}\right)\times 40x=40x$$

Since \frac{x}{40x} and \frac{40}{40x} have the same denominator, add them by adding their numerators.

$$1200+20\times \frac{x+40}{40x}\times 40x=40x$$

Multiply 20 and 40 to get 800.

$$1200+800\times \frac{x+40}{40x}x=40x$$

Express 800\times \frac{x+40}{40x} as a single fraction.

$$1200+\frac{800\left(x+40\right)}{40x}x=40x$$

Cancel out 40 in both numerator and denominator.

$$1200+\frac{20\left(x+40\right)}{x}x=40x$$

Express \frac{20\left(x+40\right)}{x}x as a single fraction.

$$1200+\frac{20\left(x+40\right)x}{x}=40x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1200 times \frac{x}{x}.

$$\frac{1200x}{x}+\frac{20\left(x+40\right)x}{x}=40x$$

Since \frac{1200x}{x} and \frac{20\left(x+40\right)x}{x} have the same denominator, add them by adding their numerators.

$$\frac{1200x+20\left(x+40\right)x}{x}=40x$$

Do the multiplications in 1200x+20\left(x+40\right)x.

$$\frac{1200x+20x^{2}+800x}{x}=40x$$

Combine like terms in 1200x+20x^{2}+800x.

$$\frac{2000x+20x^{2}}{x}=40x$$

Subtract 40x from both sides.

$$\frac{2000x+20x^{2}}{x}-40x=0$$

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

$$\frac{2000x+20x^{2}}{x}+\frac{-40xx}{x}=0$$

Since \frac{2000x+20x^{2}}{x} and \frac{-40xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{2000x+20x^{2}-40xx}{x}=0$$

Do the multiplications in 2000x+20x^{2}-40xx.

$$\frac{2000x+20x^{2}-40x^{2}}{x}=0$$

Combine like terms in 2000x+20x^{2}-40x^{2}.

$$\frac{2000x-20x^{2}}{x}=0$$

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

$$2000x-20x^{2}=0$$

Factor out x.

$$x\left(2000-20x\right)=0$$

To find equation solutions, solve x=0 and 2000-20x=0.

$$x=0 x=100$$

Variable x cannot be equal to 0.

$$x=100$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 40x, the least common multiple of x,40.

$$40\times 30+20\left(\frac{1}{40}+\frac{1}{x}\right)\times 40x=40x$$

Multiply 40 and 30 to get 1200.

$$1200+20\left(\frac{1}{40}+\frac{1}{x}\right)\times 40x=40x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 40 and x is 40x. Multiply \frac{1}{40} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{40}{40}.

$$1200+20\left(\frac{x}{40x}+\frac{40}{40x}\right)\times 40x=40x$$

Since \frac{x}{40x} and \frac{40}{40x} have the same denominator, add them by adding their numerators.

$$1200+20\times \frac{x+40}{40x}\times 40x=40x$$

Multiply 20 and 40 to get 800.

$$1200+800\times \frac{x+40}{40x}x=40x$$

Express 800\times \frac{x+40}{40x} as a single fraction.

$$1200+\frac{800\left(x+40\right)}{40x}x=40x$$

Cancel out 40 in both numerator and denominator.

$$1200+\frac{20\left(x+40\right)}{x}x=40x$$

Express \frac{20\left(x+40\right)}{x}x as a single fraction.

$$1200+\frac{20\left(x+40\right)x}{x}=40x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1200 times \frac{x}{x}.

$$\frac{1200x}{x}+\frac{20\left(x+40\right)x}{x}=40x$$

Since \frac{1200x}{x} and \frac{20\left(x+40\right)x}{x} have the same denominator, add them by adding their numerators.

$$\frac{1200x+20\left(x+40\right)x}{x}=40x$$

Do the multiplications in 1200x+20\left(x+40\right)x.

$$\frac{1200x+20x^{2}+800x}{x}=40x$$

Combine like terms in 1200x+20x^{2}+800x.

$$\frac{2000x+20x^{2}}{x}=40x$$

Subtract 40x from both sides.

$$\frac{2000x+20x^{2}}{x}-40x=0$$

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

$$\frac{2000x+20x^{2}}{x}+\frac{-40xx}{x}=0$$

Since \frac{2000x+20x^{2}}{x} and \frac{-40xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{2000x+20x^{2}-40xx}{x}=0$$

Do the multiplications in 2000x+20x^{2}-40xx.

$$\frac{2000x+20x^{2}-40x^{2}}{x}=0$$

Combine like terms in 2000x+20x^{2}-40x^{2}.

$$\frac{2000x-20x^{2}}{x}=0$$

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

$$2000x-20x^{2}=0$$

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.

$$-20x^{2}+2000x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 2000 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2000±\sqrt{2000^{2}}}{2\left(-20\right)}$$

Take the square root of 2000^{2}.

$$x=\frac{-2000±2000}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-2000±2000}{-40}$$

Now solve the equation x=\frac{-2000±2000}{-40} when ± is plus. Add -2000 to 2000.

$$x=\frac{0}{-40}$$

Divide 0 by -40.

$$x=0$$

Now solve the equation x=\frac{-2000±2000}{-40} when ± is minus. Subtract 2000 from -2000.

$$x=-\frac{4000}{-40}$$

Divide -4000 by -40.

$$x=100$$

The equation is now solved.

$$x=0 x=100$$

Variable x cannot be equal to 0.

$$x=100$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 40x, the least common multiple of x,40.

$$40\times 30+20\left(\frac{1}{40}+\frac{1}{x}\right)\times 40x=40x$$

Multiply 40 and 30 to get 1200.

$$1200+20\left(\frac{1}{40}+\frac{1}{x}\right)\times 40x=40x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 40 and x is 40x. Multiply \frac{1}{40} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{40}{40}.

$$1200+20\left(\frac{x}{40x}+\frac{40}{40x}\right)\times 40x=40x$$

Since \frac{x}{40x} and \frac{40}{40x} have the same denominator, add them by adding their numerators.

$$1200+20\times \frac{x+40}{40x}\times 40x=40x$$

Multiply 20 and 40 to get 800.

$$1200+800\times \frac{x+40}{40x}x=40x$$

Express 800\times \frac{x+40}{40x} as a single fraction.

$$1200+\frac{800\left(x+40\right)}{40x}x=40x$$

Cancel out 40 in both numerator and denominator.

$$1200+\frac{20\left(x+40\right)}{x}x=40x$$

Express \frac{20\left(x+40\right)}{x}x as a single fraction.

$$1200+\frac{20\left(x+40\right)x}{x}=40x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1200 times \frac{x}{x}.

$$\frac{1200x}{x}+\frac{20\left(x+40\right)x}{x}=40x$$

Since \frac{1200x}{x} and \frac{20\left(x+40\right)x}{x} have the same denominator, add them by adding their numerators.

$$\frac{1200x+20\left(x+40\right)x}{x}=40x$$

Do the multiplications in 1200x+20\left(x+40\right)x.

$$\frac{1200x+20x^{2}+800x}{x}=40x$$

Combine like terms in 1200x+20x^{2}+800x.

$$\frac{2000x+20x^{2}}{x}=40x$$

Subtract 40x from both sides.

$$\frac{2000x+20x^{2}}{x}-40x=0$$

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

$$\frac{2000x+20x^{2}}{x}+\frac{-40xx}{x}=0$$

Since \frac{2000x+20x^{2}}{x} and \frac{-40xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{2000x+20x^{2}-40xx}{x}=0$$

Do the multiplications in 2000x+20x^{2}-40xx.

$$\frac{2000x+20x^{2}-40x^{2}}{x}=0$$

Combine like terms in 2000x+20x^{2}-40x^{2}.

$$\frac{2000x-20x^{2}}{x}=0$$

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

$$2000x-20x^{2}=0$$

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.

$$-20x^{2}+2000x=0$$

Divide both sides by -20.

$$\frac{-20x^{2}+2000x}{-20}=\frac{0}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{2000}{-20}x=\frac{0}{-20}$$

Divide 2000 by -20.

$$x^{2}-100x=\frac{0}{-20}$$

Divide 0 by -20.

$$x^{2}-100x=0$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=2500$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=2500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2500}$$

Simplify.

$$x-50=50 x-50=-50$$

Add 50 to both sides of the equation.

$$x=100 x=0$$

Variable x cannot be equal to 0.

$$x=100$$