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

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

Solve for x

$$x=9$$

Steps Using Factoring

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

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

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

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

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

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

Use the distributive property to multiply 2x-6 by x.

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

Subtract 2x^{2} from both sides.

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

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

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

Add 6x to both sides.

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

Combine 3x and 6x to get 9x.

$$-x^{2}+9x=0$$

Factor out x.

$$x\left(-x+9\right)=0$$

To find equation solutions, solve x=0 and -x+9=0.

$$x=0 x=9$$

Variable x cannot be equal to 0.

$$x=9$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

Use the distributive property to multiply 2x-6 by x.

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

Subtract 2x^{2} from both sides.

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

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

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

Add 6x to both sides.

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

Combine 3x and 6x to get 9x.

$$-x^{2}+9x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Take the square root of 9^{2}.

$$x=\frac{-9±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±9}{-2}$$

Now solve the equation x=\frac{-9±9}{-2} when ± is plus. Add -9 to 9.

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-9±9}{-2} when ± is minus. Subtract 9 from -9.

$$x=-\frac{18}{-2}$$

Divide -18 by -2.

$$x=9$$

The equation is now solved.

$$x=0 x=9$$

Variable x cannot be equal to 0.

$$x=9$$

Steps for Completing the Square

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

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

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

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

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

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

Use the distributive property to multiply 2x-6 by x.

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

Subtract 2x^{2} from both sides.

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

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

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

Add 6x to both sides.

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

Combine 3x and 6x to get 9x.

$$-x^{2}+9x=0$$

Divide both sides by -1.

$$\frac{-x^{2}+9x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{9}{-1}x=\frac{0}{-1}$$

Divide 9 by -1.

$$x^{2}-9x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-9x=0$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=\frac{81}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{9}{2} x-\frac{9}{2}=-\frac{9}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=9 x=0$$

Variable x cannot be equal to 0.

$$x=9$$