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

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

Solve for x

$$x=-3$$

$$x=5$$

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 2\left(x-1\right)\left(x+1\right), the least common multiple of x-1,2,x+1.

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

Use the distributive property to multiply 2x+2 by 4.

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

Multiply 2 and -\frac{1}{2} to get -1.

$$8x+8-\left(x-1\right)\left(x+1\right)=\left(2x-2\right)\times 3$$

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

$$8x+8+\left(-x+1\right)\left(x+1\right)=\left(2x-2\right)\times 3$$

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

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

Add 8 and 1 to get 9.

$$8x+9-x^{2}=\left(2x-2\right)\times 3$$

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

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

Subtract 6x from both sides.

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

Combine 8x and -6x to get 2x.

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

Add 6 to both sides.

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

Add 9 and 6 to get 15.

$$2x+15-x^{2}=0$$

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.

$$-x^{2}+2x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\times 15}}{2\left(-1\right)}$$

Multiply 4 times 15.

$$x=\frac{-2±\sqrt{4+60}}{2\left(-1\right)}$$

Add 4 to 60.

$$x=\frac{-2±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$x=\frac{-2±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±8}{-2}$$

Now solve the equation x=\frac{-2±8}{-2} when ± is plus. Add -2 to 8.

$$x=\frac{6}{-2}$$

Divide 6 by -2.

$$x=-3$$

Now solve the equation x=\frac{-2±8}{-2} when ± is minus. Subtract 8 from -2.

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

Divide -10 by -2.

$$x=5$$

The equation is now solved.

$$x=-3 x=5$$

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 2\left(x-1\right)\left(x+1\right), the least common multiple of x-1,2,x+1.

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

Use the distributive property to multiply 2x+2 by 4.

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

Multiply 2 and -\frac{1}{2} to get -1.

$$8x+8-\left(x-1\right)\left(x+1\right)=\left(2x-2\right)\times 3$$

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

$$8x+8+\left(-x+1\right)\left(x+1\right)=\left(2x-2\right)\times 3$$

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

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

Add 8 and 1 to get 9.

$$8x+9-x^{2}=\left(2x-2\right)\times 3$$

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

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

Subtract 6x from both sides.

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

Combine 8x and -6x to get 2x.

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

Subtract 9 from both sides.

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

Subtract 9 from -6 to get -15.

$$2x-x^{2}=-15$$

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.

$$-x^{2}+2x=-15$$

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{15}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=-\frac{15}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=-\frac{15}{-1}$$

Divide -15 by -1.

$$x^{2}-2x=15$$

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.

$$x^{2}-2x+1=15+1$$

Add 15 to 1.

$$x^{2}-2x+1=16$$

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}.

$$\left(x-1\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-1=4 x-1=-4$$

Add 1 to both sides of the equation.

$$x=5 x=-3$$