Solve 8/x+1+6/x-1=5

$$\frac { 8 } { x + 1 } + \frac { 6 } { x - 1 } = 5$$

Solve for x

$$x=3$$

$$x=-\frac{1}{5}=-0.2$$

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.

$$\left(x-1\right)\times 8+\left(x+1\right)\times 6=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-1 by 8.

$$8x-8+\left(x+1\right)\times 6=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x+1 by 6.

$$8x-8+6x+6=5\left(x-1\right)\left(x+1\right)$$

Combine 8x and 6x to get 14x.

$$14x-8+6=5\left(x-1\right)\left(x+1\right)$$

Add -8 and 6 to get -2.

$$14x-2=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 5 by x-1.

$$14x-2=\left(5x-5\right)\left(x+1\right)$$

Use the distributive property to multiply 5x-5 by x+1 and combine like terms.

$$14x-2=5x^{2}-5$$

Subtract 5x^{2} from both sides.

$$14x-2-5x^{2}=-5$$

Add 5 to both sides.

$$14x-2-5x^{2}+5=0$$

Add -2 and 5 to get 3.

$$14x+3-5x^{2}=0$$

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.

$$-5x^{2}+14x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 14 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-14±\sqrt{196+20\times 3}}{2\left(-5\right)}$$

Multiply 20 times 3.

$$x=\frac{-14±\sqrt{196+60}}{2\left(-5\right)}$$

Add 196 to 60.

$$x=\frac{-14±\sqrt{256}}{2\left(-5\right)}$$

Take the square root of 256.

$$x=\frac{-14±16}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-14±16}{-10}$$

Now solve the equation x=\frac{-14±16}{-10} when ± is plus. Add -14 to 16.

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

Reduce the fraction \frac{2}{-10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{5}$$

Now solve the equation x=\frac{-14±16}{-10} when ± is minus. Subtract 16 from -14.

$$x=-\frac{30}{-10}$$

Divide -30 by -10.

$$x=3$$

The equation is now solved.

$$x=-\frac{1}{5} x=3$$

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.

$$\left(x-1\right)\times 8+\left(x+1\right)\times 6=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-1 by 8.

$$8x-8+\left(x+1\right)\times 6=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x+1 by 6.

$$8x-8+6x+6=5\left(x-1\right)\left(x+1\right)$$

Combine 8x and 6x to get 14x.

$$14x-8+6=5\left(x-1\right)\left(x+1\right)$$

Add -8 and 6 to get -2.

$$14x-2=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 5 by x-1.

$$14x-2=\left(5x-5\right)\left(x+1\right)$$

Use the distributive property to multiply 5x-5 by x+1 and combine like terms.

$$14x-2=5x^{2}-5$$

Subtract 5x^{2} from both sides.

$$14x-2-5x^{2}=-5$$

Add 2 to both sides.

$$14x-5x^{2}=-5+2$$

Add -5 and 2 to get -3.

$$14x-5x^{2}=-3$$

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.

$$-5x^{2}+14x=-3$$

Divide both sides by -5.

$$\frac{-5x^{2}+14x}{-5}=-\frac{3}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{14}{-5}x=-\frac{3}{-5}$$

Divide 14 by -5.

$$x^{2}-\frac{14}{5}x=-\frac{3}{-5}$$

Divide -3 by -5.

$$x^{2}-\frac{14}{5}x=\frac{3}{5}$$

Divide -\frac{14}{5}, the coefficient of the x term, by 2 to get -\frac{7}{5}. Then add the square of -\frac{7}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14}{5}x+\left(-\frac{7}{5}\right)^{2}=\frac{3}{5}+\left(-\frac{7}{5}\right)^{2}$$

Square -\frac{7}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14}{5}x+\frac{49}{25}=\frac{3}{5}+\frac{49}{25}$$

Add \frac{3}{5} to \frac{49}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{14}{5}x+\frac{49}{25}=\frac{64}{25}$$

Factor x^{2}-\frac{14}{5}x+\frac{49}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{5}\right)^{2}=\frac{64}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{5}\right)^{2}}=\sqrt{\frac{64}{25}}$$

Simplify.

$$x-\frac{7}{5}=\frac{8}{5} x-\frac{7}{5}=-\frac{8}{5}$$

Add \frac{7}{5} to both sides of the equation.

$$x=3 x=-\frac{1}{5}$$