Solve 5/x-1+2/x+1=-6

$$\frac { 5 } { x - 1 } + \frac { 2 } { x + 1 } = - 6$$

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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

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.

$$\left(x+1\right)\times 5+\left(x-1\right)\times 2=-6\left(x-1\right)\left(x+1\right)$$

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

$$5x+5+\left(x-1\right)\times 2=-6\left(x-1\right)\left(x+1\right)$$

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

$$5x+5+2x-2=-6\left(x-1\right)\left(x+1\right)$$

Combine 5x and 2x to get 7x.

$$7x+5-2=-6\left(x-1\right)\left(x+1\right)$$

Subtract 2 from 5 to get 3.

$$7x+3=-6\left(x-1\right)\left(x+1\right)$$

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

$$7x+3=\left(-6x+6\right)\left(x+1\right)$$

Use the distributive property to multiply -6x+6 by x+1 and combine like terms.

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

Add 6x^{2} to both sides.

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

Subtract 6 from both sides.

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

Subtract 6 from 3 to get -3.

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

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.

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

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

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

Square 7.

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

Multiply -4 times 6.

$$x=\frac{-7±\sqrt{49-24\left(-3\right)}}{2\times 6}$$

Multiply -24 times -3.

$$x=\frac{-7±\sqrt{49+72}}{2\times 6}$$

Add 49 to 72.

$$x=\frac{-7±\sqrt{121}}{2\times 6}$$

Take the square root of 121.

$$x=\frac{-7±11}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-7±11}{12}$$

Now solve the equation x=\frac{-7±11}{12} when ± is plus. Add -7 to 11.

$$x=\frac{4}{12}$$

Reduce the fraction \frac{4}{12} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{-7±11}{12} when ± is minus. Subtract 11 from -7.

$$x=-\frac{18}{12}$$

Reduce the fraction \frac{-18}{12} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

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

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.

$$\left(x+1\right)\times 5+\left(x-1\right)\times 2=-6\left(x-1\right)\left(x+1\right)$$

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

$$5x+5+\left(x-1\right)\times 2=-6\left(x-1\right)\left(x+1\right)$$

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

$$5x+5+2x-2=-6\left(x-1\right)\left(x+1\right)$$

Combine 5x and 2x to get 7x.

$$7x+5-2=-6\left(x-1\right)\left(x+1\right)$$

Subtract 2 from 5 to get 3.

$$7x+3=-6\left(x-1\right)\left(x+1\right)$$

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

$$7x+3=\left(-6x+6\right)\left(x+1\right)$$

Use the distributive property to multiply -6x+6 by x+1 and combine like terms.

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

Add 6x^{2} to both sides.

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

Subtract 3 from both sides.

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

Subtract 3 from 6 to get 3.

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

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.

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

Divide both sides by 6.

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

Dividing by 6 undoes the multiplication by 6.

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

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{7}{6}x=\frac{1}{2}$$

Divide \frac{7}{6}, the coefficient of the x term, by 2 to get \frac{7}{12}. Then add the square of \frac{7}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{6}x+\left(\frac{7}{12}\right)^{2}=\frac{1}{2}+\left(\frac{7}{12}\right)^{2}$$

Square \frac{7}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{1}{2}+\frac{49}{144}$$

Add \frac{1}{2} to \frac{49}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{121}{144}$$

Factor x^{2}+\frac{7}{6}x+\frac{49}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{12}\right)^{2}=\frac{121}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{12}\right)^{2}}=\sqrt{\frac{121}{144}}$$

Simplify.

$$x+\frac{7}{12}=\frac{11}{12} x+\frac{7}{12}=-\frac{11}{12}$$

Subtract \frac{7}{12} from both sides of the equation.

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