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

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

Solve for x

$$x=\frac{\sqrt{145}-11}{2}\approx 0.520797289$$

$$x=\frac{-\sqrt{145}-11}{2}\approx -11.520797289$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right)\left(x+2\right), the least common multiple of x-1,x+2,x.

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

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

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

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

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

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

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

Use the distributive property to multiply x^{2}-x by x-2 and combine like terms.

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

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

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

Combine 3x^{2} and -3x^{2} to get 0.

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

Combine 2x and 2x to get 4x.

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

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

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

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

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

Use the distributive property to multiply x^{3}+x^{2}-2x by 4.

$$2x^{3}+4x=4x^{3}+4x^{2}-8x-\left(x-1\right)\left(x+2\right)\left(2x+3\right)$$

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

$$2x^{3}+4x=4x^{3}+4x^{2}-8x-\left(x^{2}+x-2\right)\left(2x+3\right)$$

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

$$2x^{3}+4x=4x^{3}+4x^{2}-8x-\left(2x^{3}+5x^{2}-x-6\right)$$

To find the opposite of 2x^{3}+5x^{2}-x-6, find the opposite of each term.

$$2x^{3}+4x=4x^{3}+4x^{2}-8x-2x^{3}-5x^{2}+x+6$$

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

$$2x^{3}+4x=2x^{3}+4x^{2}-8x-5x^{2}+x+6$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

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

Combine -8x and x to get -7x.

$$2x^{3}+4x=2x^{3}-x^{2}-7x+6$$

Subtract 2x^{3} from both sides.

$$2x^{3}+4x-2x^{3}=-x^{2}-7x+6$$

Combine 2x^{3} and -2x^{3} to get 0.

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

Add x^{2} to both sides.

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

Add 7x to both sides.

$$4x+x^{2}+7x=6$$

Combine 4x and 7x to get 11x.

$$11x+x^{2}=6$$

Subtract 6 from both sides.

$$11x+x^{2}-6=0$$

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.

$$x^{2}+11x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 11 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-6\right)}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{-11±\sqrt{121+24}}{2}$$

Add 121 to 24.

$$x=\frac{-11±\sqrt{145}}{2}$$

Now solve the equation x=\frac{-11±\sqrt{145}}{2} when ± is plus. Add -11 to \sqrt{145}.

$$x=\frac{\sqrt{145}-11}{2}$$

Now solve the equation x=\frac{-11±\sqrt{145}}{2} when ± is minus. Subtract \sqrt{145} from -11.

$$x=\frac{-\sqrt{145}-11}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{145}-11}{2} x=\frac{-\sqrt{145}-11}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right)\left(x+2\right), the least common multiple of x-1,x+2,x.

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

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

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

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

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

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

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

Use the distributive property to multiply x^{2}-x by x-2 and combine like terms.

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

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

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

Combine 3x^{2} and -3x^{2} to get 0.

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

Combine 2x and 2x to get 4x.

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

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

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

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

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

Use the distributive property to multiply x^{3}+x^{2}-2x by 4.

$$2x^{3}+4x=4x^{3}+4x^{2}-8x-\left(x-1\right)\left(x+2\right)\left(2x+3\right)$$

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

$$2x^{3}+4x=4x^{3}+4x^{2}-8x-\left(x^{2}+x-2\right)\left(2x+3\right)$$

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

$$2x^{3}+4x=4x^{3}+4x^{2}-8x-\left(2x^{3}+5x^{2}-x-6\right)$$

To find the opposite of 2x^{3}+5x^{2}-x-6, find the opposite of each term.

$$2x^{3}+4x=4x^{3}+4x^{2}-8x-2x^{3}-5x^{2}+x+6$$

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

$$2x^{3}+4x=2x^{3}+4x^{2}-8x-5x^{2}+x+6$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

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

Combine -8x and x to get -7x.

$$2x^{3}+4x=2x^{3}-x^{2}-7x+6$$

Subtract 2x^{3} from both sides.

$$2x^{3}+4x-2x^{3}=-x^{2}-7x+6$$

Combine 2x^{3} and -2x^{3} to get 0.

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

Add x^{2} to both sides.

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

Add 7x to both sides.

$$4x+x^{2}+7x=6$$

Combine 4x and 7x to get 11x.

$$11x+x^{2}=6$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+11x=6$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=6+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=6+\frac{121}{4}$$

Add 6 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{145}{4}$$

Factor x^{2}+11x+\frac{121}{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}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{145}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{145}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{\sqrt{145}}{2} x+\frac{11}{2}=-\frac{\sqrt{145}}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=\frac{\sqrt{145}-11}{2} x=\frac{-\sqrt{145}-11}{2}$$