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

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

Solve for x

$$x=\sqrt{11}+3\approx 6.31662479$$

$$x=3-\sqrt{11}\approx -0.31662479$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{2}{3},3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(3x+2\right), the least common multiple of 3x+2,3-x.

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

Use the distributive property to multiply x-3 by x.

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

Use the distributive property to multiply -2-3x by -1.

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

Subtract 2 from both sides.

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

Subtract 3x from both sides.

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

Combine -3x and -3x to get -6x.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$x=\frac{-\left(-6\right)±\sqrt{36+8}}{2}$$

Add 36 to 8.

$$x=\frac{-\left(-6\right)±\sqrt{44}}{2}$$

Take the square root of 44.

$$x=\frac{-\left(-6\right)±2\sqrt{11}}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{11}}{2}$$

Now solve the equation x=\frac{6±2\sqrt{11}}{2} when ± is plus. Add 6 to 2\sqrt{11}.

$$x=\frac{2\sqrt{11}+6}{2}$$

Divide 6+2\sqrt{11} by 2.

$$x=\sqrt{11}+3$$

Now solve the equation x=\frac{6±2\sqrt{11}}{2} when ± is minus. Subtract 2\sqrt{11} from 6.

$$x=\frac{6-2\sqrt{11}}{2}$$

Divide 6-2\sqrt{11} by 2.

$$x=3-\sqrt{11}$$

The equation is now solved.

$$x=\sqrt{11}+3 x=3-\sqrt{11}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{2}{3},3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(3x+2\right), the least common multiple of 3x+2,3-x.

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

Use the distributive property to multiply x-3 by x.

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

Use the distributive property to multiply -2-3x by -1.

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

Subtract 3x from both sides.

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

Combine -3x and -3x to get -6x.

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

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

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

Square -3.

$$x^{2}-6x+9=2+9$$

Add 2 to 9.

$$x^{2}-6x+9=11$$

Factor x^{2}-6x+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}.

$$\left(x-3\right)^{2}=11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{11}$$

Simplify.

$$x-3=\sqrt{11} x-3=-\sqrt{11}$$

Add 3 to both sides of the equation.

$$x=\sqrt{11}+3 x=3-\sqrt{11}$$