Solve x+1/x-2=x+5/x-3+1

$$\frac { x + 1 } { x - 2 } = \frac { x + 5 } { x - 3 } + 1$$

Solve for x

$$x=1$$

$$x=-1$$

Steps by Finding Square Root

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

$$\left(x-3\right)\left(x+1\right)=\left(x-2\right)\left(x+5\right)+\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-3 by x+1 and combine like terms.

$$x^{2}-2x-3=\left(x-2\right)\left(x+5\right)+\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-2 by x+5 and combine like terms.

$$x^{2}-2x-3=x^{2}+3x-10+\left(x-3\right)\left(x-2\right)$$

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

$$x^{2}-2x-3=x^{2}+3x-10+x^{2}-5x+6$$

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

$$x^{2}-2x-3=2x^{2}+3x-10-5x+6$$

Combine 3x and -5x to get -2x.

$$x^{2}-2x-3=2x^{2}-2x-10+6$$

Add -10 and 6 to get -4.

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

Subtract 2x^{2} from both sides.

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

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

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

Add 2x to both sides.

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

Combine -2x and 2x to get 0.

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

Add 3 to both sides.

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

Add -4 and 3 to get -1.

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

Divide both sides by -1.

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

Divide -1 by -1 to get 1.

$$x^{2}=1$$

Take the square root of both sides of the equation.

$$x=1 x=-1$$

Steps Using the Quadratic Formula

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

$$\left(x-3\right)\left(x+1\right)=\left(x-2\right)\left(x+5\right)+\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-3 by x+1 and combine like terms.

$$x^{2}-2x-3=\left(x-2\right)\left(x+5\right)+\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-2 by x+5 and combine like terms.

$$x^{2}-2x-3=x^{2}+3x-10+\left(x-3\right)\left(x-2\right)$$

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

$$x^{2}-2x-3=x^{2}+3x-10+x^{2}-5x+6$$

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

$$x^{2}-2x-3=2x^{2}+3x-10-5x+6$$

Combine 3x and -5x to get -2x.

$$x^{2}-2x-3=2x^{2}-2x-10+6$$

Add -10 and 6 to get -4.

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

Subtract 2x^{2} from both sides.

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

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

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

Add 2x to both sides.

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

Combine -2x and 2x to get 0.

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

Add 4 to both sides.

$$-x^{2}-3+4=0$$

Add -3 and 4 to get 1.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4}}{2\left(-1\right)}$$

Take the square root of 4.

$$x=\frac{0±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±2}{-2}$$

Now solve the equation x=\frac{0±2}{-2} when ± is plus. Divide 2 by -2.

$$x=-1$$

Now solve the equation x=\frac{0±2}{-2} when ± is minus. Divide -2 by -2.

$$x=1$$

The equation is now solved.

$$x=-1 x=1$$