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

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{23}i-3}{8}\approx -0.375-0.59947894i$$

$$x=\frac{-3+\sqrt{23}i}{8}\approx -0.375+0.59947894i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+1\right), the least common multiple of x+1,3x.

$$-3xx=\left(x+1\right)\left(x+2\right)$$

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

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

Use the distributive property to multiply x+1 by x+2 and combine like terms.

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

Subtract x^{2} from both sides.

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

Subtract 3x from both sides.

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

Subtract 2 from both sides.

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

Combine -3x^{2} and -x^{2} to get -4x^{2}.

$$-4x^{2}-3x-2=0$$

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

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-4\right)\left(-2\right)}}{2\left(-4\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-4\right)\left(-2\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-3\right)±\sqrt{9+16\left(-2\right)}}{2\left(-4\right)}$$

Multiply 16 times -2.

$$x=\frac{-\left(-3\right)±\sqrt{9-32}}{2\left(-4\right)}$$

Add 9 to -32.

$$x=\frac{-\left(-3\right)±\sqrt{-23}}{2\left(-4\right)}$$

Take the square root of -23.

$$x=\frac{-\left(-3\right)±\sqrt{23}i}{2\left(-4\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{23}i}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{3±\sqrt{23}i}{-8}$$

Now solve the equation x=\frac{3±\sqrt{23}i}{-8} when ± is plus. Add 3 to i\sqrt{23}.

$$x=\frac{3+\sqrt{23}i}{-8}$$

Divide 3+i\sqrt{23} by -8.

$$x=\frac{-\sqrt{23}i-3}{8}$$

Now solve the equation x=\frac{3±\sqrt{23}i}{-8} when ± is minus. Subtract i\sqrt{23} from 3.

$$x=\frac{-\sqrt{23}i+3}{-8}$$

Divide 3-i\sqrt{23} by -8.

$$x=\frac{-3+\sqrt{23}i}{8}$$

The equation is now solved.

$$x=\frac{-\sqrt{23}i-3}{8} x=\frac{-3+\sqrt{23}i}{8}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+1\right), the least common multiple of x+1,3x.

$$-3xx=\left(x+1\right)\left(x+2\right)$$

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

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

Use the distributive property to multiply x+1 by x+2 and combine like terms.

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

Subtract x^{2} from both sides.

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

Subtract 3x from both sides.

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

Combine -3x^{2} and -x^{2} to get -4x^{2}.

$$-4x^{2}-3x=2$$

Divide both sides by -4.

$$\frac{-4x^{2}-3x}{-4}=\frac{2}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{3}{-4}\right)x=\frac{2}{-4}$$

Divide -3 by -4.

$$x^{2}+\frac{3}{4}x=\frac{2}{-4}$$

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

$$x^{2}+\frac{3}{4}x=-\frac{1}{2}$$

Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=-\frac{1}{2}+\left(\frac{3}{8}\right)^{2}$$

Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{1}{2}+\frac{9}{64}$$

Add -\frac{1}{2} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{23}{64}$$

Factor x^{2}+\frac{3}{4}x+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{8}\right)^{2}=-\frac{23}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{-\frac{23}{64}}$$

Simplify.

$$x+\frac{3}{8}=\frac{\sqrt{23}i}{8} x+\frac{3}{8}=-\frac{\sqrt{23}i}{8}$$

Subtract \frac{3}{8} from both sides of the equation.

$$x=\frac{-3+\sqrt{23}i}{8} x=\frac{-\sqrt{23}i-3}{8}$$