Solve 0=3x-1+1/x+1

$$0 = 3 x - 1 + \frac { 1 } { x + 1 }$$

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$0=3x\left(x+1\right)+\left(x+1\right)\left(-1\right)+1$$

Use the distributive property to multiply 3x by x+1.

$$0=3x^{2}+3x+\left(x+1\right)\left(-1\right)+1$$

Use the distributive property to multiply x+1 by -1.

$$0=3x^{2}+3x-x-1+1$$

Combine 3x and -x to get 2x.

$$0=3x^{2}+2x-1+1$$

Add -1 and 1 to get 0.

$$0=3x^{2}+2x$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+2x=0$$

Factor out x.

$$x\left(3x+2\right)=0$$

To find equation solutions, solve x=0 and 3x+2=0.

$$x=0 x=-\frac{2}{3}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$0=3x\left(x+1\right)+\left(x+1\right)\left(-1\right)+1$$

Use the distributive property to multiply 3x by x+1.

$$0=3x^{2}+3x+\left(x+1\right)\left(-1\right)+1$$

Use the distributive property to multiply x+1 by -1.

$$0=3x^{2}+3x-x-1+1$$

Combine 3x and -x to get 2x.

$$0=3x^{2}+2x-1+1$$

Add -1 and 1 to get 0.

$$0=3x^{2}+2x$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+2x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}}}{2\times 3}$$

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-2±2}{6}$$

Now solve the equation x=\frac{-2±2}{6} when ± is plus. Add -2 to 2.

$$x=\frac{0}{6}$$

Divide 0 by 6.

$$x=0$$

Now solve the equation x=\frac{-2±2}{6} when ± is minus. Subtract 2 from -2.

$$x=-\frac{4}{6}$$

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{3}$$

The equation is now solved.

$$x=0 x=-\frac{2}{3}$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$0=3x\left(x+1\right)+\left(x+1\right)\left(-1\right)+1$$

Use the distributive property to multiply 3x by x+1.

$$0=3x^{2}+3x+\left(x+1\right)\left(-1\right)+1$$

Use the distributive property to multiply x+1 by -1.

$$0=3x^{2}+3x-x-1+1$$

Combine 3x and -x to get 2x.

$$0=3x^{2}+2x-1+1$$

Add -1 and 1 to get 0.

$$0=3x^{2}+2x$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+2x=0$$

Divide both sides by 3.

$$\frac{3x^{2}+2x}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{2}{3}x=\frac{0}{3}$$

Divide 0 by 3.

$$x^{2}+\frac{2}{3}x=0$$

Divide \frac{2}{3}, the coefficient of the x term, by 2 to get \frac{1}{3}. Then add the square of \frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.

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

Factor x^{2}+\frac{2}{3}x+\frac{1}{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}.

$$\left(x+\frac{1}{3}\right)^{2}=\frac{1}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Simplify.

$$x+\frac{1}{3}=\frac{1}{3} x+\frac{1}{3}=-\frac{1}{3}$$

Subtract \frac{1}{3} from both sides of the equation.

$$x=0 x=-\frac{2}{3}$$