Solve x-2/x-1+x/x+3=0

$$\frac { x - 2 } { x - 1 } + \frac { x } { x + 3 } = 0$$

Solve for x

$$x=\sqrt{3}\approx 1.732050808$$

$$x=-\sqrt{3}\approx -1.732050808$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -3,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+3\right), the least common multiple of x-1,x+3.

$$\left(x+3\right)\left(x-2\right)+\left(x-1\right)x=0$$

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

$$x^{2}+x-6+\left(x-1\right)x=0$$

Use the distributive property to multiply x-1 by x.

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

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

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

Combine x and -x to get 0.

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

Add 6 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2.

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

Divide 6 by 2 to get 3.

$$x^{2}=3$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+3\right), the least common multiple of x-1,x+3.

$$\left(x+3\right)\left(x-2\right)+\left(x-1\right)x=0$$

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

$$x^{2}+x-6+\left(x-1\right)x=0$$

Use the distributive property to multiply x-1 by x.

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

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

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

Combine x and -x to get 0.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\times 2\left(-6\right)}}{2\times 2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 2\left(-6\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{0±\sqrt{-8\left(-6\right)}}{2\times 2}$$

Multiply -8 times -6.

$$x=\frac{0±\sqrt{48}}{2\times 2}$$

Take the square root of 48.

$$x=\frac{0±4\sqrt{3}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{0±4\sqrt{3}}{4}$$

Now solve the equation x=\frac{0±4\sqrt{3}}{4} when ± is plus.

$$x=\sqrt{3}$$

Now solve the equation x=\frac{0±4\sqrt{3}}{4} when ± is minus.

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

The equation is now solved.

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