Solve 2/n+1+3/2(n-2)=23/5n

$$\frac { 2 } { n + 1 } + \frac { 3 } { 2 ( n - 2 ) } = \frac { 23 } { 5 n }$$

Solve for n

$$n = -\frac{23}{11} = -2\frac{1}{11} \approx -2.090909091$$

$$n=4$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -1,0,2 since division by zero is not defined. Multiply both sides of the equation by 10n\left(n-2\right)\left(n+1\right), the least common multiple of n+1,2\left(n-2\right),5n.

$$10n\left(n-2\right)\times 2+5n\left(n+1\right)\times 3=2\left(n-2\right)\left(n+1\right)\times 23$$

Use the distributive property to multiply 10n by n-2.

$$\left(10n^{2}-20n\right)\times 2+5n\left(n+1\right)\times 3=2\left(n-2\right)\left(n+1\right)\times 23$$

Use the distributive property to multiply 10n^{2}-20n by 2.

$$20n^{2}-40n+5n\left(n+1\right)\times 3=2\left(n-2\right)\left(n+1\right)\times 23$$

Multiply 5 and 3 to get 15.

$$20n^{2}-40n+15n\left(n+1\right)=2\left(n-2\right)\left(n+1\right)\times 23$$

Use the distributive property to multiply 15n by n+1.

$$20n^{2}-40n+15n^{2}+15n=2\left(n-2\right)\left(n+1\right)\times 23$$

Combine 20n^{2} and 15n^{2} to get 35n^{2}.

$$35n^{2}-40n+15n=2\left(n-2\right)\left(n+1\right)\times 23$$

Combine -40n and 15n to get -25n.

$$35n^{2}-25n=2\left(n-2\right)\left(n+1\right)\times 23$$

Multiply 2 and 23 to get 46.

$$35n^{2}-25n=46\left(n-2\right)\left(n+1\right)$$

Use the distributive property to multiply 46 by n-2.

$$35n^{2}-25n=\left(46n-92\right)\left(n+1\right)$$

Use the distributive property to multiply 46n-92 by n+1 and combine like terms.

$$35n^{2}-25n=46n^{2}-46n-92$$

Subtract 46n^{2} from both sides.

$$35n^{2}-25n-46n^{2}=-46n-92$$

Combine 35n^{2} and -46n^{2} to get -11n^{2}.

$$-11n^{2}-25n=-46n-92$$

Add 46n to both sides.

$$-11n^{2}-25n+46n=-92$$

Combine -25n and 46n to get 21n.

$$-11n^{2}+21n=-92$$

Add 92 to both sides.

$$-11n^{2}+21n+92=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -11 for a, 21 for b, and 92 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-21±\sqrt{21^{2}-4\left(-11\right)\times 92}}{2\left(-11\right)}$$

Square 21.

$$n=\frac{-21±\sqrt{441-4\left(-11\right)\times 92}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$n=\frac{-21±\sqrt{441+44\times 92}}{2\left(-11\right)}$$

Multiply 44 times 92.

$$n=\frac{-21±\sqrt{441+4048}}{2\left(-11\right)}$$

Add 441 to 4048.

$$n=\frac{-21±\sqrt{4489}}{2\left(-11\right)}$$

Take the square root of 4489.

$$n=\frac{-21±67}{2\left(-11\right)}$$

Multiply 2 times -11.

$$n=\frac{-21±67}{-22}$$

Now solve the equation n=\frac{-21±67}{-22} when ± is plus. Add -21 to 67.

$$n=\frac{46}{-22}$$

Reduce the fraction \frac{46}{-22} to lowest terms by extracting and canceling out 2.

$$n=-\frac{23}{11}$$

Now solve the equation n=\frac{-21±67}{-22} when ± is minus. Subtract 67 from -21.

$$n=-\frac{88}{-22}$$

Divide -88 by -22.

$$n=4$$

The equation is now solved.

$$n=-\frac{23}{11} n=4$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -1,0,2 since division by zero is not defined. Multiply both sides of the equation by 10n\left(n-2\right)\left(n+1\right), the least common multiple of n+1,2\left(n-2\right),5n.

$$10n\left(n-2\right)\times 2+5n\left(n+1\right)\times 3=2\left(n-2\right)\left(n+1\right)\times 23$$

Use the distributive property to multiply 10n by n-2.

$$\left(10n^{2}-20n\right)\times 2+5n\left(n+1\right)\times 3=2\left(n-2\right)\left(n+1\right)\times 23$$

Use the distributive property to multiply 10n^{2}-20n by 2.

$$20n^{2}-40n+5n\left(n+1\right)\times 3=2\left(n-2\right)\left(n+1\right)\times 23$$

Multiply 5 and 3 to get 15.

$$20n^{2}-40n+15n\left(n+1\right)=2\left(n-2\right)\left(n+1\right)\times 23$$

Use the distributive property to multiply 15n by n+1.

$$20n^{2}-40n+15n^{2}+15n=2\left(n-2\right)\left(n+1\right)\times 23$$

Combine 20n^{2} and 15n^{2} to get 35n^{2}.

$$35n^{2}-40n+15n=2\left(n-2\right)\left(n+1\right)\times 23$$

Combine -40n and 15n to get -25n.

$$35n^{2}-25n=2\left(n-2\right)\left(n+1\right)\times 23$$

Multiply 2 and 23 to get 46.

$$35n^{2}-25n=46\left(n-2\right)\left(n+1\right)$$

Use the distributive property to multiply 46 by n-2.

$$35n^{2}-25n=\left(46n-92\right)\left(n+1\right)$$

Use the distributive property to multiply 46n-92 by n+1 and combine like terms.

$$35n^{2}-25n=46n^{2}-46n-92$$

Subtract 46n^{2} from both sides.

$$35n^{2}-25n-46n^{2}=-46n-92$$

Combine 35n^{2} and -46n^{2} to get -11n^{2}.

$$-11n^{2}-25n=-46n-92$$

Add 46n to both sides.

$$-11n^{2}-25n+46n=-92$$

Combine -25n and 46n to get 21n.

$$-11n^{2}+21n=-92$$

Divide both sides by -11.

$$\frac{-11n^{2}+21n}{-11}=-\frac{92}{-11}$$

Dividing by -11 undoes the multiplication by -11.

$$n^{2}+\frac{21}{-11}n=-\frac{92}{-11}$$

Divide 21 by -11.

$$n^{2}-\frac{21}{11}n=-\frac{92}{-11}$$

Divide -92 by -11.

$$n^{2}-\frac{21}{11}n=\frac{92}{11}$$

Divide -\frac{21}{11}, the coefficient of the x term, by 2 to get -\frac{21}{22}. Then add the square of -\frac{21}{22} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{21}{11}n+\left(-\frac{21}{22}\right)^{2}=\frac{92}{11}+\left(-\frac{21}{22}\right)^{2}$$

Square -\frac{21}{22} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{21}{11}n+\frac{441}{484}=\frac{92}{11}+\frac{441}{484}$$

Add \frac{92}{11} to \frac{441}{484} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{21}{11}n+\frac{441}{484}=\frac{4489}{484}$$

Factor n^{2}-\frac{21}{11}n+\frac{441}{484}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{21}{22}\right)^{2}=\frac{4489}{484}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{21}{22}\right)^{2}}=\sqrt{\frac{4489}{484}}$$

Simplify.

$$n-\frac{21}{22}=\frac{67}{22} n-\frac{21}{22}=-\frac{67}{22}$$

Add \frac{21}{22} to both sides of the equation.

$$n=4 n=-\frac{23}{11}$$