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

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

Solve for x

$$x=\frac{\sqrt{37}-9}{2}\approx -1.458618735$$

$$x=\frac{-\sqrt{37}-9}{2}\approx -7.541381265$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

To find the opposite of 2x^{2}+6x+4, find the opposite of each term.

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

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

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

Subtract 4 from -1 to get -5.

$$-x^{2}-5-6x=\left(x+2\right)\times 3$$

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

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

Subtract 3x from both sides.

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

Combine -6x and -3x to get -9x.

$$-x^{2}-5-9x=6$$

Subtract 6 from both sides.

$$-x^{2}-5-9x-6=0$$

Subtract 6 from -5 to get -11.

$$-x^{2}-11-9x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-9x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -9 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-1\right)\left(-11\right)}}{2\left(-1\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-1\right)\left(-11\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-9\right)±\sqrt{81+4\left(-11\right)}}{2\left(-1\right)}$$

Multiply 4 times -11.

$$x=\frac{-\left(-9\right)±\sqrt{81-44}}{2\left(-1\right)}$$

Add 81 to -44.

$$x=\frac{-\left(-9\right)±\sqrt{37}}{2\left(-1\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{37}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{9±\sqrt{37}}{-2}$$

Now solve the equation x=\frac{9±\sqrt{37}}{-2} when ± is plus. Add 9 to \sqrt{37}.

$$x=\frac{\sqrt{37}+9}{-2}$$

Divide 9+\sqrt{37} by -2.

$$x=\frac{-\sqrt{37}-9}{2}$$

Now solve the equation x=\frac{9±\sqrt{37}}{-2} when ± is minus. Subtract \sqrt{37} from 9.

$$x=\frac{9-\sqrt{37}}{-2}$$

Divide 9-\sqrt{37} by -2.

$$x=\frac{\sqrt{37}-9}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{37}-9}{2} x=\frac{\sqrt{37}-9}{2}$$

Steps for Completing the Square

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

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

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

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

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

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

To find the opposite of 2x^{2}+6x+4, find the opposite of each term.

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

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

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

Subtract 4 from -1 to get -5.

$$-x^{2}-5-6x=\left(x+2\right)\times 3$$

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

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

Subtract 3x from both sides.

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

Combine -6x and -3x to get -9x.

$$-x^{2}-5-9x=6$$

Add 5 to both sides.

$$-x^{2}-9x=6+5$$

Add 6 and 5 to get 11.

$$-x^{2}-9x=11$$

Divide both sides by -1.

$$\frac{-x^{2}-9x}{-1}=\frac{11}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{9}{-1}\right)x=\frac{11}{-1}$$

Divide -9 by -1.

$$x^{2}+9x=\frac{11}{-1}$$

Divide 11 by -1.

$$x^{2}+9x=-11$$

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.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-11+\left(\frac{9}{2}\right)^{2}$$

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

$$x^{2}+9x+\frac{81}{4}=-11+\frac{81}{4}$$

Add -11 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{37}{4}$$

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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{37}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{\sqrt{37}}{2} x+\frac{9}{2}=-\frac{\sqrt{37}}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=\frac{\sqrt{37}-9}{2} x=\frac{-\sqrt{37}-9}{2}$$