Solve (x+1)*2/x-6/60=2

$$(x+1) \times \frac{ \frac{ 2 }{ x } -6 }{ 60 } =2$$

Solve for x

$$x=\frac{2\sqrt{241}-31}{3}\approx 0.016116464$$

$$x=\frac{-2\sqrt{241}-31}{3}\approx -20.682783131$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 60.

$$\left(x+1\right)\left(\frac{2}{x}-6\right)=120$$

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

$$\left(x+1\right)\left(\frac{2}{x}-\frac{6x}{x}\right)=120$$

Since \frac{2}{x} and \frac{6x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\left(x+1\right)\times \frac{2-6x}{x}=120$$

Express \left(x+1\right)\times \frac{2-6x}{x} as a single fraction.

$$\frac{\left(x+1\right)\left(2-6x\right)}{x}=120$$

Apply the distributive property by multiplying each term of x+1 by each term of 2-6x.

$$\frac{2x-6x^{2}+2-6x}{x}=120$$

Combine 2x and -6x to get -4x.

$$\frac{-4x-6x^{2}+2}{x}=120$$

Subtract 120 from both sides.

$$\frac{-4x-6x^{2}+2}{x}-120=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 120 times \frac{x}{x}.

$$\frac{-4x-6x^{2}+2}{x}-\frac{120x}{x}=0$$

Since \frac{-4x-6x^{2}+2}{x} and \frac{120x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x-6x^{2}+2-120x}{x}=0$$

Combine like terms in -4x-6x^{2}+2-120x.

$$\frac{-124x-6x^{2}+2}{x}=0$$

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

$$-124x-6x^{2}+2=0$$

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.

$$-6x^{2}-124x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, -124 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-124\right)±\sqrt{\left(-124\right)^{2}-4\left(-6\right)\times 2}}{2\left(-6\right)}$$

Square -124.

$$x=\frac{-\left(-124\right)±\sqrt{15376-4\left(-6\right)\times 2}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-\left(-124\right)±\sqrt{15376+24\times 2}}{2\left(-6\right)}$$

Multiply 24 times 2.

$$x=\frac{-\left(-124\right)±\sqrt{15376+48}}{2\left(-6\right)}$$

Add 15376 to 48.

$$x=\frac{-\left(-124\right)±\sqrt{15424}}{2\left(-6\right)}$$

Take the square root of 15424.

$$x=\frac{-\left(-124\right)±8\sqrt{241}}{2\left(-6\right)}$$

The opposite of -124 is 124.

$$x=\frac{124±8\sqrt{241}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{124±8\sqrt{241}}{-12}$$

Now solve the equation x=\frac{124±8\sqrt{241}}{-12} when ± is plus. Add 124 to 8\sqrt{241}.

$$x=\frac{8\sqrt{241}+124}{-12}$$

Divide 124+8\sqrt{241} by -12.

$$x=\frac{-2\sqrt{241}-31}{3}$$

Now solve the equation x=\frac{124±8\sqrt{241}}{-12} when ± is minus. Subtract 8\sqrt{241} from 124.

$$x=\frac{124-8\sqrt{241}}{-12}$$

Divide 124-8\sqrt{241} by -12.

$$x=\frac{2\sqrt{241}-31}{3}$$

The equation is now solved.

$$x=\frac{-2\sqrt{241}-31}{3} x=\frac{2\sqrt{241}-31}{3}$$

Steps for Completing the Square

Multiply both sides of the equation by 60.

$$\left(x+1\right)\left(\frac{2}{x}-6\right)=120$$

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

$$\left(x+1\right)\left(\frac{2}{x}-\frac{6x}{x}\right)=120$$

Since \frac{2}{x} and \frac{6x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\left(x+1\right)\times \frac{2-6x}{x}=120$$

Express \left(x+1\right)\times \frac{2-6x}{x} as a single fraction.

$$\frac{\left(x+1\right)\left(2-6x\right)}{x}=120$$

Apply the distributive property by multiplying each term of x+1 by each term of 2-6x.

$$\frac{2x-6x^{2}+2-6x}{x}=120$$

Combine 2x and -6x to get -4x.

$$\frac{-4x-6x^{2}+2}{x}=120$$

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

$$-4x-6x^{2}+2=120x$$

Subtract 120x from both sides.

$$-4x-6x^{2}+2-120x=0$$

Combine -4x and -120x to get -124x.

$$-124x-6x^{2}+2=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$-124x-6x^{2}=-2$$

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.

$$-6x^{2}-124x=-2$$

Divide both sides by -6.

$$\frac{-6x^{2}-124x}{-6}=-\frac{2}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\left(-\frac{124}{-6}\right)x=-\frac{2}{-6}$$

Reduce the fraction \frac{-124}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{62}{3}x=-\frac{2}{-6}$$

Reduce the fraction \frac{-2}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{62}{3}x=\frac{1}{3}$$

Divide \frac{62}{3}, the coefficient of the x term, by 2 to get \frac{31}{3}. Then add the square of \frac{31}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{62}{3}x+\left(\frac{31}{3}\right)^{2}=\frac{1}{3}+\left(\frac{31}{3}\right)^{2}$$

Square \frac{31}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{62}{3}x+\frac{961}{9}=\frac{1}{3}+\frac{961}{9}$$

Add \frac{1}{3} to \frac{961}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{62}{3}x+\frac{961}{9}=\frac{964}{9}$$

Factor x^{2}+\frac{62}{3}x+\frac{961}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{31}{3}\right)^{2}=\frac{964}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{31}{3}\right)^{2}}=\sqrt{\frac{964}{9}}$$

Simplify.

$$x+\frac{31}{3}=\frac{2\sqrt{241}}{3} x+\frac{31}{3}=-\frac{2\sqrt{241}}{3}$$

Subtract \frac{31}{3} from both sides of the equation.

$$x=\frac{2\sqrt{241}-31}{3} x=\frac{-2\sqrt{241}-31}{3}$$