Question 151730
{{{(n)/(n-3)+n=(7n-15)/(n-3)}}} Start with the given equation.



{{{cross((n-3))((n)/cross((n-3)))+(n-3)(n)=cross((n-3))((7n-15)/cross(n-3))}}} Multiply <font size="4"><b>every</b></font> term on both sides by the LCD {{{n-3}}}. Doing this will eliminate all of the fractions.




{{{n+n(n-3)=7n-15}}} Simplify



{{{n+n^2-3n=7n-15}}} Distribute



{{{n+n^2-3n-7n+15=0}}} Subtract {{{7n}}} from both sides. Add 15 to both sides.



{{{n^2-9n+15=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{an^2+bn+c}}} where {{{a=1}}}, {{{b=-9}}}, and {{{c=15}}}



Let's use the quadratic formula to solve for n



{{{n = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{n = (-(-9) +- sqrt( (-9)^2-4(1)(15) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-9}}}, and {{{c=15}}}



{{{n = (9 +- sqrt( (-9)^2-4(1)(15) ))/(2(1))}}} Negate {{{-9}}} to get {{{9}}}. 



{{{n = (9 +- sqrt( 81-4(1)(15) ))/(2(1))}}} Square {{{-9}}} to get {{{81}}}. 



{{{n = (9 +- sqrt( 81-60 ))/(2(1))}}} Multiply {{{4(1)(15)}}} to get {{{60}}}



{{{n = (9 +- sqrt( 21 ))/(2(1))}}} Subtract {{{60}}} from {{{81}}} to get {{{21}}}



{{{n = (9 +- sqrt( 21 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{n = (9+sqrt(21))/(2)}}} or {{{n = (9-sqrt(21))/(2)}}} Break up the expression.  



So our answers are {{{n = (9+sqrt(21))/(2)}}} or {{{n = (9-sqrt(21))/(2)}}} 



which approximate to {{{n=6.791}}} or {{{n=2.209}}}