Solve 3/x+1+4/x-1=1/x-2

$$\frac { 3 } { x + 1 } + \frac { 4 } { x - 1 } = \frac { 1 } { x - 2 }$$

Solve for x

$$x = \frac{\sqrt{193} + 13}{12} \approx 2.241036999$$

$$x=\frac{13-\sqrt{193}}{12}\approx -0.074370332$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(x+1\right), the least common multiple of x+1,x-1,x-2.

$$\left(x-2\right)\left(x-1\right)\times 3+\left(x-2\right)\left(x+1\right)\times 4=x^{2}-1$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\left(x^{2}-3x+2\right)\times 3+\left(x-2\right)\left(x+1\right)\times 4=x^{2}-1$$

Use the distributive property to multiply x^{2}-3x+2 by 3.

$$3x^{2}-9x+6+\left(x-2\right)\left(x+1\right)\times 4=x^{2}-1$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$3x^{2}-9x+6+\left(x^{2}-x-2\right)\times 4=x^{2}-1$$

Use the distributive property to multiply x^{2}-x-2 by 4.

$$3x^{2}-9x+6+4x^{2}-4x-8=x^{2}-1$$

Combine 3x^{2} and 4x^{2} to get 7x^{2}.

$$7x^{2}-9x+6-4x-8=x^{2}-1$$

Combine -9x and -4x to get -13x.

$$7x^{2}-13x+6-8=x^{2}-1$$

Subtract 8 from 6 to get -2.

$$7x^{2}-13x-2=x^{2}-1$$

Subtract x^{2} from both sides.

$$7x^{2}-13x-2-x^{2}=-1$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-13x-2=-1$$

Add 1 to both sides.

$$6x^{2}-13x-2+1=0$$

Add -2 and 1 to get -1.

$$6x^{2}-13x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -13 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 6\left(-1\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-13\right)±\sqrt{169-24\left(-1\right)}}{2\times 6}$$

Multiply -24 times -1.

$$x=\frac{-\left(-13\right)±\sqrt{169+24}}{2\times 6}$$

Add 169 to 24.

$$x=\frac{-\left(-13\right)±\sqrt{193}}{2\times 6}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{193}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{13±\sqrt{193}}{12}$$

Now solve the equation x=\frac{13±\sqrt{193}}{12} when ± is plus. Add 13 to \sqrt{193}.

$$x=\frac{\sqrt{193}+13}{12}$$

Now solve the equation x=\frac{13±\sqrt{193}}{12} when ± is minus. Subtract \sqrt{193} from 13.

$$x=\frac{13-\sqrt{193}}{12}$$

The equation is now solved.

$$x=\frac{\sqrt{193}+13}{12} x=\frac{13-\sqrt{193}}{12}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(x+1\right), the least common multiple of x+1,x-1,x-2.

$$\left(x-2\right)\left(x-1\right)\times 3+\left(x-2\right)\left(x+1\right)\times 4=x^{2}-1$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\left(x^{2}-3x+2\right)\times 3+\left(x-2\right)\left(x+1\right)\times 4=x^{2}-1$$

Use the distributive property to multiply x^{2}-3x+2 by 3.

$$3x^{2}-9x+6+\left(x-2\right)\left(x+1\right)\times 4=x^{2}-1$$

Use the distributive property to multiply x-2 by x+1 and combine like terms.

$$3x^{2}-9x+6+\left(x^{2}-x-2\right)\times 4=x^{2}-1$$

Use the distributive property to multiply x^{2}-x-2 by 4.

$$3x^{2}-9x+6+4x^{2}-4x-8=x^{2}-1$$

Combine 3x^{2} and 4x^{2} to get 7x^{2}.

$$7x^{2}-9x+6-4x-8=x^{2}-1$$

Combine -9x and -4x to get -13x.

$$7x^{2}-13x+6-8=x^{2}-1$$

Subtract 8 from 6 to get -2.

$$7x^{2}-13x-2=x^{2}-1$$

Subtract x^{2} from both sides.

$$7x^{2}-13x-2-x^{2}=-1$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-13x-2=-1$$

Add 2 to both sides.

$$6x^{2}-13x=-1+2$$

Add -1 and 2 to get 1.

$$6x^{2}-13x=1$$

Divide both sides by 6.

$$\frac{6x^{2}-13x}{6}=\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{13}{6}x=\frac{1}{6}$$

Divide -\frac{13}{6}, the coefficient of the x term, by 2 to get -\frac{13}{12}. Then add the square of -\frac{13}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=\frac{1}{6}+\left(-\frac{13}{12}\right)^{2}$$

Square -\frac{13}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{1}{6}+\frac{169}{144}$$

Add \frac{1}{6} to \frac{169}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{193}{144}$$

Factor x^{2}-\frac{13}{6}x+\frac{169}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{193}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{193}{144}}$$

Simplify.

$$x-\frac{13}{12}=\frac{\sqrt{193}}{12} x-\frac{13}{12}=-\frac{\sqrt{193}}{12}$$

Add \frac{13}{12} to both sides of the equation.

$$x=\frac{\sqrt{193}+13}{12} x=\frac{13-\sqrt{193}}{12}$$