Solve 3x/x+2=3/x

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

Solve for x

$$x=-1$$

$$x=2$$

Steps Using the Quadratic Formula

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

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

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

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

Use the distributive property to multiply x+2 by 3.

$$x^{2}\times 3=3x+6$$

Subtract 3x from both sides.

$$x^{2}\times 3-3x=6$$

Subtract 6 from both sides.

$$x^{2}\times 3-3x-6=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$3x^{2}-3x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -3 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 3\left(-6\right)}}{2\times 3}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 3\left(-6\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-3\right)±\sqrt{9-12\left(-6\right)}}{2\times 3}$$

Multiply -12 times -6.

$$x=\frac{-\left(-3\right)±\sqrt{9+72}}{2\times 3}$$

Add 9 to 72.

$$x=\frac{-\left(-3\right)±\sqrt{81}}{2\times 3}$$

Take the square root of 81.

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

The opposite of -3 is 3.

$$x=\frac{3±9}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{3±9}{6}$$

Now solve the equation x=\frac{3±9}{6} when ± is plus. Add 3 to 9.

$$x=\frac{12}{6}$$

Divide 12 by 6.

$$x=2$$

Now solve the equation x=\frac{3±9}{6} when ± is minus. Subtract 9 from 3.

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

Divide -6 by 6.

$$x=-1$$

The equation is now solved.

$$x=2 x=-1$$

Steps for Completing the Square

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

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

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

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

Use the distributive property to multiply x+2 by 3.

$$x^{2}\times 3=3x+6$$

Subtract 3x from both sides.

$$x^{2}\times 3-3x=6$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$3x^{2}-3x=6$$

Divide both sides by 3.

$$\frac{3x^{2}-3x}{3}=\frac{6}{3}$$

Dividing by 3 undoes the multiplication by 3.

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

Divide -3 by 3.

$$x^{2}-x=\frac{6}{3}$$

Divide 6 by 3.

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

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

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add 2 to \frac{1}{4}.

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

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

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

Add \frac{1}{2} to both sides of the equation.

$$x=2 x=-1$$