Solve 5x/x+1-2/x-1=8

$$\frac { 5 x } { x + 1 } - \frac { 2 } { x - 1 } = 8$$

Solve for x

$$x=\frac{2}{3}\approx 0.666666667$$

$$x=-3$$

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.

$$\left(x-1\right)\times 5x-\left(x+1\right)\times 2=8\left(x-1\right)\left(x+1\right)$$

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

$$\left(5x-5\right)x-\left(x+1\right)\times 2=8\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-5x-\left(x+1\right)\times 2=8\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-5x-\left(2x+2\right)=8\left(x-1\right)\left(x+1\right)$$

To find the opposite of 2x+2, find the opposite of each term.

$$5x^{2}-5x-2x-2=8\left(x-1\right)\left(x+1\right)$$

Combine -5x and -2x to get -7x.

$$5x^{2}-7x-2=8\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-7x-2=\left(8x-8\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-8 by x+1 and combine like terms.

$$5x^{2}-7x-2=8x^{2}-8$$

Subtract 8x^{2} from both sides.

$$5x^{2}-7x-2-8x^{2}=-8$$

Combine 5x^{2} and -8x^{2} to get -3x^{2}.

$$-3x^{2}-7x-2=-8$$

Add 8 to both sides.

$$-3x^{2}-7x-2+8=0$$

Add -2 and 8 to get 6.

$$-3x^{2}-7x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -7 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\times 6}}{2\left(-3\right)}$$

Multiply 12 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49+72}}{2\left(-3\right)}$$

Add 49 to 72.

$$x=\frac{-\left(-7\right)±\sqrt{121}}{2\left(-3\right)}$$

Take the square root of 121.

$$x=\frac{-\left(-7\right)±11}{2\left(-3\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±11}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{7±11}{-6}$$

Now solve the equation x=\frac{7±11}{-6} when ± is plus. Add 7 to 11.

$$x=\frac{18}{-6}$$

Divide 18 by -6.

$$x=-3$$

Now solve the equation x=\frac{7±11}{-6} when ± is minus. Subtract 11 from 7.

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

Reduce the fraction \frac{-4}{-6} to lowest terms by extracting and canceling out 2.

$$x=\frac{2}{3}$$

The equation is now solved.

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

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.

$$\left(x-1\right)\times 5x-\left(x+1\right)\times 2=8\left(x-1\right)\left(x+1\right)$$

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

$$\left(5x-5\right)x-\left(x+1\right)\times 2=8\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-5x-\left(x+1\right)\times 2=8\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-5x-\left(2x+2\right)=8\left(x-1\right)\left(x+1\right)$$

To find the opposite of 2x+2, find the opposite of each term.

$$5x^{2}-5x-2x-2=8\left(x-1\right)\left(x+1\right)$$

Combine -5x and -2x to get -7x.

$$5x^{2}-7x-2=8\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-7x-2=\left(8x-8\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-8 by x+1 and combine like terms.

$$5x^{2}-7x-2=8x^{2}-8$$

Subtract 8x^{2} from both sides.

$$5x^{2}-7x-2-8x^{2}=-8$$

Combine 5x^{2} and -8x^{2} to get -3x^{2}.

$$-3x^{2}-7x-2=-8$$

Add 2 to both sides.

$$-3x^{2}-7x=-8+2$$

Add -8 and 2 to get -6.

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

Divide both sides by -3.

$$\frac{-3x^{2}-7x}{-3}=-\frac{6}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{7}{-3}\right)x=-\frac{6}{-3}$$

Divide -7 by -3.

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

Divide -6 by -3.

$$x^{2}+\frac{7}{3}x=2$$

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=2+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=2+\frac{49}{36}$$

Add 2 to \frac{49}{36}.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{121}{36}$$

Factor x^{2}+\frac{7}{3}x+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{121}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Simplify.

$$x+\frac{7}{6}=\frac{11}{6} x+\frac{7}{6}=-\frac{11}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

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