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

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

Solve for x

$$x=\sqrt{13}+3\approx 6.605551275$$

$$x=3-\sqrt{13}\approx -0.605551275$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{2}{3},0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(3x+2\right), the least common multiple of 3x+2,2x.

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

Use the distributive property to multiply 2x by x+1.

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

Use the distributive property to multiply 3x+2 by x-2 and combine like terms.

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

Subtract 3x^{2} from both sides.

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

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

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

Add 4x to both sides.

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

Combine 2x and 4x to get 6x.

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

Add 4 to both sides.

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

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

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\times 4}}{2\left(-1\right)}$$

Multiply 4 times 4.

$$x=\frac{-6±\sqrt{36+16}}{2\left(-1\right)}$$

Add 36 to 16.

$$x=\frac{-6±\sqrt{52}}{2\left(-1\right)}$$

Take the square root of 52.

$$x=\frac{-6±2\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±2\sqrt{13}}{-2}$$

Now solve the equation x=\frac{-6±2\sqrt{13}}{-2} when ± is plus. Add -6 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}-6}{-2}$$

Divide -6+2\sqrt{13} by -2.

$$x=3-\sqrt{13}$$

Now solve the equation x=\frac{-6±2\sqrt{13}}{-2} when ± is minus. Subtract 2\sqrt{13} from -6.

$$x=\frac{-2\sqrt{13}-6}{-2}$$

Divide -6-2\sqrt{13} by -2.

$$x=\sqrt{13}+3$$

The equation is now solved.

$$x=3-\sqrt{13} x=\sqrt{13}+3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{2}{3},0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(3x+2\right), the least common multiple of 3x+2,2x.

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

Use the distributive property to multiply 2x by x+1.

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

Use the distributive property to multiply 3x+2 by x-2 and combine like terms.

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

Subtract 3x^{2} from both sides.

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

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

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

Add 4x to both sides.

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

Combine 2x and 4x to get 6x.

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

Divide both sides by -1.

$$\frac{-x^{2}+6x}{-1}=-\frac{4}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{6}{-1}x=-\frac{4}{-1}$$

Divide 6 by -1.

$$x^{2}-6x=-\frac{4}{-1}$$

Divide -4 by -1.

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

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.

$$x^{2}-6x+\left(-3\right)^{2}=4+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=4+9$$

Add 4 to 9.

$$x^{2}-6x+9=13$$

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}.

$$\left(x-3\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{13}$$

Simplify.

$$x-3=\sqrt{13} x-3=-\sqrt{13}$$

Add 3 to both sides of the equation.

$$x=\sqrt{13}+3 x=3-\sqrt{13}$$