Solve 28/x-1+16/x+1=3

$$\frac{ 28 }{ x-1 } + \frac{ 16 }{ x+1 } =3$$

Solve for x

$$x=15$$

$$x=-\frac{1}{3}\approx -0.333333333$$

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.

$$\left(x+1\right)\times 28+\left(x-1\right)\times 16=3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x+1 by 28.

$$28x+28+\left(x-1\right)\times 16=3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-1 by 16.

$$28x+28+16x-16=3\left(x-1\right)\left(x+1\right)$$

Combine 28x and 16x to get 44x.

$$44x+28-16=3\left(x-1\right)\left(x+1\right)$$

Subtract 16 from 28 to get 12.

$$44x+12=3\left(x-1\right)\left(x+1\right)$$

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

$$44x+12=\left(3x-3\right)\left(x+1\right)$$

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

$$44x+12=3x^{2}-3$$

Subtract 3x^{2} from both sides.

$$44x+12-3x^{2}=-3$$

Add 3 to both sides.

$$44x+12-3x^{2}+3=0$$

Add 12 and 3 to get 15.

$$44x+15-3x^{2}=0$$

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.

$$-3x^{2}+44x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 44 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-44±\sqrt{44^{2}-4\left(-3\right)\times 15}}{2\left(-3\right)}$$

Square 44.

$$x=\frac{-44±\sqrt{1936-4\left(-3\right)\times 15}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-44±\sqrt{1936+12\times 15}}{2\left(-3\right)}$$

Multiply 12 times 15.

$$x=\frac{-44±\sqrt{1936+180}}{2\left(-3\right)}$$

Add 1936 to 180.

$$x=\frac{-44±\sqrt{2116}}{2\left(-3\right)}$$

Take the square root of 2116.

$$x=\frac{-44±46}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-44±46}{-6}$$

Now solve the equation x=\frac{-44±46}{-6} when ± is plus. Add -44 to 46.

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

Reduce the fraction \frac{2}{-6} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-44±46}{-6} when ± is minus. Subtract 46 from -44.

$$x=-\frac{90}{-6}$$

Divide -90 by -6.

$$x=15$$

The equation is now solved.

$$x=-\frac{1}{3} x=15$$

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.

$$\left(x+1\right)\times 28+\left(x-1\right)\times 16=3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x+1 by 28.

$$28x+28+\left(x-1\right)\times 16=3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-1 by 16.

$$28x+28+16x-16=3\left(x-1\right)\left(x+1\right)$$

Combine 28x and 16x to get 44x.

$$44x+28-16=3\left(x-1\right)\left(x+1\right)$$

Subtract 16 from 28 to get 12.

$$44x+12=3\left(x-1\right)\left(x+1\right)$$

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

$$44x+12=\left(3x-3\right)\left(x+1\right)$$

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

$$44x+12=3x^{2}-3$$

Subtract 3x^{2} from both sides.

$$44x+12-3x^{2}=-3$$

Subtract 12 from both sides.

$$44x-3x^{2}=-3-12$$

Subtract 12 from -3 to get -15.

$$44x-3x^{2}=-15$$

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.

$$-3x^{2}+44x=-15$$

Divide both sides by -3.

$$\frac{-3x^{2}+44x}{-3}=-\frac{15}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{44}{-3}x=-\frac{15}{-3}$$

Divide 44 by -3.

$$x^{2}-\frac{44}{3}x=-\frac{15}{-3}$$

Divide -15 by -3.

$$x^{2}-\frac{44}{3}x=5$$

Divide -\frac{44}{3}, the coefficient of the x term, by 2 to get -\frac{22}{3}. Then add the square of -\frac{22}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{44}{3}x+\left(-\frac{22}{3}\right)^{2}=5+\left(-\frac{22}{3}\right)^{2}$$

Square -\frac{22}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{44}{3}x+\frac{484}{9}=5+\frac{484}{9}$$

Add 5 to \frac{484}{9}.

$$x^{2}-\frac{44}{3}x+\frac{484}{9}=\frac{529}{9}$$

Factor x^{2}-\frac{44}{3}x+\frac{484}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{22}{3}\right)^{2}=\frac{529}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{22}{3}\right)^{2}}=\sqrt{\frac{529}{9}}$$

Simplify.

$$x-\frac{22}{3}=\frac{23}{3} x-\frac{22}{3}=-\frac{23}{3}$$

Add \frac{22}{3} to both sides of the equation.

$$x=15 x=-\frac{1}{3}$$