Solve 5/u-1=-9+2/u+1

$$\frac { 5 } { u - 1 } = - 9 + \frac { 2 } { u + 1 }$$

Solve for u

$$u=-\frac{2}{3}\approx -0.666666667$$

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

Steps Using the Quadratic Formula

Variable u 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(u-1\right)\left(u+1\right), the least common multiple of u-1,u+1.

$$\left(u+1\right)\times 5=\left(u-1\right)\left(u+1\right)\left(-9\right)+\left(u-1\right)\times 2$$

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

$$5u+5=\left(u-1\right)\left(u+1\right)\left(-9\right)+\left(u-1\right)\times 2$$

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

$$5u+5=\left(u^{2}-1\right)\left(-9\right)+\left(u-1\right)\times 2$$

Use the distributive property to multiply u^{2}-1 by -9.

$$5u+5=-9u^{2}+9+\left(u-1\right)\times 2$$

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

$$5u+5=-9u^{2}+9+2u-2$$

Subtract 2 from 9 to get 7.

$$5u+5=-9u^{2}+7+2u$$

Add 9u^{2} to both sides.

$$5u+5+9u^{2}=7+2u$$

Subtract 7 from both sides.

$$5u+5+9u^{2}-7=2u$$

Subtract 7 from 5 to get -2.

$$5u-2+9u^{2}=2u$$

Subtract 2u from both sides.

$$5u-2+9u^{2}-2u=0$$

Combine 5u and -2u to get 3u.

$$3u-2+9u^{2}=0$$

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.

$$9u^{2}+3u-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 3 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-3±\sqrt{3^{2}-4\times 9\left(-2\right)}}{2\times 9}$$

Square 3.

$$u=\frac{-3±\sqrt{9-4\times 9\left(-2\right)}}{2\times 9}$$

Multiply -4 times 9.

$$u=\frac{-3±\sqrt{9-36\left(-2\right)}}{2\times 9}$$

Multiply -36 times -2.

$$u=\frac{-3±\sqrt{9+72}}{2\times 9}$$

Add 9 to 72.

$$u=\frac{-3±\sqrt{81}}{2\times 9}$$

Take the square root of 81.

$$u=\frac{-3±9}{2\times 9}$$

Multiply 2 times 9.

$$u=\frac{-3±9}{18}$$

Now solve the equation u=\frac{-3±9}{18} when ± is plus. Add -3 to 9.

$$u=\frac{6}{18}$$

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

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

Now solve the equation u=\frac{-3±9}{18} when ± is minus. Subtract 9 from -3.

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Variable u 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(u-1\right)\left(u+1\right), the least common multiple of u-1,u+1.

$$\left(u+1\right)\times 5=\left(u-1\right)\left(u+1\right)\left(-9\right)+\left(u-1\right)\times 2$$

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

$$5u+5=\left(u-1\right)\left(u+1\right)\left(-9\right)+\left(u-1\right)\times 2$$

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

$$5u+5=\left(u^{2}-1\right)\left(-9\right)+\left(u-1\right)\times 2$$

Use the distributive property to multiply u^{2}-1 by -9.

$$5u+5=-9u^{2}+9+\left(u-1\right)\times 2$$

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

$$5u+5=-9u^{2}+9+2u-2$$

Subtract 2 from 9 to get 7.

$$5u+5=-9u^{2}+7+2u$$

Add 9u^{2} to both sides.

$$5u+5+9u^{2}=7+2u$$

Subtract 2u from both sides.

$$5u+5+9u^{2}-2u=7$$

Combine 5u and -2u to get 3u.

$$3u+5+9u^{2}=7$$

Subtract 5 from both sides.

$$3u+9u^{2}=7-5$$

Subtract 5 from 7 to get 2.

$$3u+9u^{2}=2$$

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.

$$9u^{2}+3u=2$$

Divide both sides by 9.

$$\frac{9u^{2}+3u}{9}=\frac{2}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$u^{2}+\frac{3}{9}u=\frac{2}{9}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

$$u^{2}+\frac{1}{3}u=\frac{2}{9}$$

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}+\frac{1}{3}u+\left(\frac{1}{6}\right)^{2}=\frac{2}{9}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$u^{2}+\frac{1}{3}u+\frac{1}{36}=\frac{2}{9}+\frac{1}{36}$$

Add \frac{2}{9} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$u^{2}+\frac{1}{3}u+\frac{1}{36}=\frac{1}{4}$$

Factor u^{2}+\frac{1}{3}u+\frac{1}{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}.

$$\left(u+\frac{1}{6}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u+\frac{1}{6}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$u+\frac{1}{6}=\frac{1}{2} u+\frac{1}{6}=-\frac{1}{2}$$

Subtract \frac{1}{6} from both sides of the equation.

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