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

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

Solve for x

$$x=\frac{2\sqrt{6}}{3}+1\approx 2.632993162$$

$$x=-\frac{2\sqrt{6}}{3}+1\approx -0.632993162$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply x+1 by 4.

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

Use the distributive property to multiply x-1 by 2.

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

Combine 4x and 2x to get 6x.

$$6x+4-2=3\left(x-1\right)\left(x+1\right)$$

Subtract 2 from 4 to get 2.

$$6x+2=3\left(x-1\right)\left(x+1\right)$$

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

$$6x+2=\left(3x-3\right)\left(x+1\right)$$

Use the distributive property to multiply 3x-3 by x+1 and combine like terms.

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

Subtract 3x^{2} from both sides.

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

Add 3 to both sides.

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

Add 2 and 3 to get 5.

$$6x+5-3x^{2}=0$$

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.

$$-3x^{2}+6x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 6 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-6±\sqrt{36+12\times 5}}{2\left(-3\right)}$$

Multiply 12 times 5.

$$x=\frac{-6±\sqrt{36+60}}{2\left(-3\right)}$$

Add 36 to 60.

$$x=\frac{-6±\sqrt{96}}{2\left(-3\right)}$$

Take the square root of 96.

$$x=\frac{-6±4\sqrt{6}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-6±4\sqrt{6}}{-6}$$

Now solve the equation x=\frac{-6±4\sqrt{6}}{-6} when ± is plus. Add -6 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}-6}{-6}$$

Divide -6+4\sqrt{6} by -6.

$$x=-\frac{2\sqrt{6}}{3}+1$$

Now solve the equation x=\frac{-6±4\sqrt{6}}{-6} when ± is minus. Subtract 4\sqrt{6} from -6.

$$x=\frac{-4\sqrt{6}-6}{-6}$$

Divide -6-4\sqrt{6} by -6.

$$x=\frac{2\sqrt{6}}{3}+1$$

The equation is now solved.

$$x=-\frac{2\sqrt{6}}{3}+1 x=\frac{2\sqrt{6}}{3}+1$$

Steps for Completing the Square

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

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

Use the distributive property to multiply x+1 by 4.

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

Use the distributive property to multiply x-1 by 2.

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

Combine 4x and 2x to get 6x.

$$6x+4-2=3\left(x-1\right)\left(x+1\right)$$

Subtract 2 from 4 to get 2.

$$6x+2=3\left(x-1\right)\left(x+1\right)$$

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

$$6x+2=\left(3x-3\right)\left(x+1\right)$$

Use the distributive property to multiply 3x-3 by x+1 and combine like terms.

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

Subtract 3x^{2} from both sides.

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

Subtract 2 from both sides.

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

Subtract 2 from -3 to get -5.

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

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.

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

Divide both sides by -3.

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

Dividing by -3 undoes the multiplication by -3.

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

Divide 6 by -3.

$$x^{2}-2x=-\frac{5}{-3}$$

Divide -5 by -3.

$$x^{2}-2x=\frac{5}{3}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{5}{3}+1$$

Add \frac{5}{3} to 1.

$$x^{2}-2x+1=\frac{8}{3}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=\frac{2\sqrt{6}}{3} x-1=-\frac{2\sqrt{6}}{3}$$

Add 1 to both sides of the equation.

$$x=\frac{2\sqrt{6}}{3}+1 x=-\frac{2\sqrt{6}}{3}+1$$