Solve x-2/x-3=x+2/x-6+1

$$\frac { x - 2 } { x - 3 } = \frac { x + 2 } { x - 6 } + 1$$

Solve for x

$$x=0$$

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 3,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x-3\right), the least common multiple of x-3,x-6.

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

Use the distributive property to multiply x-6 by x-2 and combine like terms.

$$x^{2}-8x+12=\left(x-3\right)\left(x+2\right)+\left(x-6\right)\left(x-3\right)$$

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

$$x^{2}-8x+12=x^{2}-x-6+\left(x-6\right)\left(x-3\right)$$

Use the distributive property to multiply x-6 by x-3 and combine like terms.

$$x^{2}-8x+12=x^{2}-x-6+x^{2}-9x+18$$

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

$$x^{2}-8x+12=2x^{2}-x-6-9x+18$$

Combine -x and -9x to get -10x.

$$x^{2}-8x+12=2x^{2}-10x-6+18$$

Add -6 and 18 to get 12.

$$x^{2}-8x+12=2x^{2}-10x+12$$

Subtract 2x^{2} from both sides.

$$x^{2}-8x+12-2x^{2}=-10x+12$$

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

$$-x^{2}-8x+12=-10x+12$$

Add 10x to both sides.

$$-x^{2}-8x+12+10x=12$$

Combine -8x and 10x to get 2x.

$$-x^{2}+2x+12=12$$

Subtract 12 from both sides.

$$-x^{2}+2x+12-12=0$$

Subtract 12 from 12 to get 0.

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

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

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

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±2}{-2}$$

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

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

Divide 0 by -2.

$$x=0$$

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

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

Divide -4 by -2.

$$x=2$$

The equation is now solved.

$$x=0 x=2$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 3,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x-3\right), the least common multiple of x-3,x-6.

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

Use the distributive property to multiply x-6 by x-2 and combine like terms.

$$x^{2}-8x+12=\left(x-3\right)\left(x+2\right)+\left(x-6\right)\left(x-3\right)$$

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

$$x^{2}-8x+12=x^{2}-x-6+\left(x-6\right)\left(x-3\right)$$

Use the distributive property to multiply x-6 by x-3 and combine like terms.

$$x^{2}-8x+12=x^{2}-x-6+x^{2}-9x+18$$

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

$$x^{2}-8x+12=2x^{2}-x-6-9x+18$$

Combine -x and -9x to get -10x.

$$x^{2}-8x+12=2x^{2}-10x-6+18$$

Add -6 and 18 to get 12.

$$x^{2}-8x+12=2x^{2}-10x+12$$

Subtract 2x^{2} from both sides.

$$x^{2}-8x+12-2x^{2}=-10x+12$$

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

$$-x^{2}-8x+12=-10x+12$$

Add 10x to both sides.

$$-x^{2}-8x+12+10x=12$$

Combine -8x and 10x to get 2x.

$$-x^{2}+2x+12=12$$

Subtract 12 from both sides.

$$-x^{2}+2x=12-12$$

Subtract 12 from 12 to get 0.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 2 by -1.

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

Divide 0 by -1.

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

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

$$x^{2}-2x+1=1$$

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

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=1 x-1=-1$$

Add 1 to both sides of the equation.

$$x=2 x=0$$