Solve x^2/x+1=-1/3

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

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{11}i}{6}\approx -0.166666667+0.552770798i$$

$$x=\frac{-\sqrt{11}i-1}{6}\approx -0.166666667-0.552770798i$$

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 3\left(x+1\right), the least common multiple of x+1,3.

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

To find the opposite of x+1, find the opposite of each term.

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

Add x to both sides.

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

Add 1 to both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 1 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 3}}{2\times 3}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-1±\sqrt{1-12}}{2\times 3}$$

Add 1 to -12.

$$x=\frac{-1±\sqrt{-11}}{2\times 3}$$

Take the square root of -11.

$$x=\frac{-1±\sqrt{11}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-1±\sqrt{11}i}{6}$$

Now solve the equation x=\frac{-1±\sqrt{11}i}{6} when ± is plus. Add -1 to i\sqrt{11}.

$$x=\frac{-1+\sqrt{11}i}{6}$$

Now solve the equation x=\frac{-1±\sqrt{11}i}{6} when ± is minus. Subtract i\sqrt{11} from -1.

$$x=\frac{-\sqrt{11}i-1}{6}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{11}i}{6} x=\frac{-\sqrt{11}i-1}{6}$$

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 3\left(x+1\right), the least common multiple of x+1,3.

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

To find the opposite of x+1, find the opposite of each term.

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

Add x to both sides.

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

Divide both sides by 3.

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

Dividing by 3 undoes the multiplication by 3.

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

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=-\frac{1}{3}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{1}{3}+\frac{1}{36}$$

Add -\frac{1}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{11}{36}$$

Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=-\frac{11}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{-\frac{11}{36}}$$

Simplify.

$$x+\frac{1}{6}=\frac{\sqrt{11}i}{6} x+\frac{1}{6}=-\frac{\sqrt{11}i}{6}$$

Subtract \frac{1}{6} from both sides of the equation.

$$x=\frac{-1+\sqrt{11}i}{6} x=\frac{-\sqrt{11}i-1}{6}$$