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

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

Solve for x

$$x=-5$$

$$x=2$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

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

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

Combine 3x and -3x to get 0.

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

Add 2 and 2 to get 4.

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

Use the distributive property to multiply 3 by x-1.

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

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

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

Subtract 3x^{2} from both sides.

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

Combine 2x^{2} and -3x^{2} to get -x^{2}.

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

Subtract 3x from both sides.

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

Add 6 to both sides.

$$-x^{2}+4-3x+6=0$$

Add 4 and 6 to get 10.

$$-x^{2}+10-3x=0$$

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.

$$-x^{2}-3x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -3 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\times 10}}{2\left(-1\right)}$$

Multiply 4 times 10.

$$x=\frac{-\left(-3\right)±\sqrt{9+40}}{2\left(-1\right)}$$

Add 9 to 40.

$$x=\frac{-\left(-3\right)±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-\left(-3\right)±7}{2\left(-1\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{3±7}{-2}$$

Now solve the equation x=\frac{3±7}{-2} when ± is plus. Add 3 to 7.

$$x=\frac{10}{-2}$$

Divide 10 by -2.

$$x=-5$$

Now solve the equation x=\frac{3±7}{-2} when ± is minus. Subtract 7 from 3.

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

Divide -4 by -2.

$$x=2$$

The equation is now solved.

$$x=-5 x=2$$

Steps for Completing the Square

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

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

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

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

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

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

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

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

Combine 3x and -3x to get 0.

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

Add 2 and 2 to get 4.

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

Use the distributive property to multiply 3 by x-1.

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

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

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

Subtract 3x^{2} from both sides.

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

Combine 2x^{2} and -3x^{2} to get -x^{2}.

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

Subtract 3x from both sides.

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

Subtract 4 from both sides.

$$-x^{2}-3x=-6-4$$

Subtract 4 from -6 to get -10.

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

Divide both sides by -1.

$$\frac{-x^{2}-3x}{-1}=-\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{3}{-1}\right)x=-\frac{10}{-1}$$

Divide -3 by -1.

$$x^{2}+3x=-\frac{10}{-1}$$

Divide -10 by -1.

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

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=10+\frac{9}{4}$$

Add 10 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{49}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{7}{2} x+\frac{3}{2}=-\frac{7}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=2 x=-5$$