Question 195568


{{{x/9-8/x=1/9}}} Start with the given equation.



{{{cross(9)x(x/cross(9))-9*cross(x)(8/cross(x))=cross(9)x(1/cross(9))}}} Multiply <font size="4"><b>every</b></font> term on both sides by the LCD {{{9x}}}. Doing this will eliminate all of the fractions.



{{{x(x)-9(8)=x(1)}}} Cancel out and simplify



{{{x^2-72=x}}} Multiply



{{{x^2-72-x=0}}} Subtract x from both sides.



{{{x^2-x-72=0}}} Rearrange the terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c=0}}} where {{{a=1}}}, {{{b=-1}}}, and {{{c=-72}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-1) +- sqrt( (-1)^2-4(1)(-72) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-1}}}, and {{{c=-72}}}



{{{x = (1 +- sqrt( (-1)^2-4(1)(-72) ))/(2(1))}}} Negate {{{-1}}} to get {{{1}}}. 



{{{x = (1 +- sqrt( 1-4(1)(-72) ))/(2(1))}}} Square {{{-1}}} to get {{{1}}}. 



{{{x = (1 +- sqrt( 1--288 ))/(2(1))}}} Multiply {{{4(1)(-72)}}} to get {{{-288}}}



{{{x = (1 +- sqrt( 1+288 ))/(2(1))}}} Rewrite {{{sqrt(1--288)}}} as {{{sqrt(1+288)}}}



{{{x = (1 +- sqrt( 289 ))/(2(1))}}} Add {{{1}}} to {{{288}}} to get {{{289}}}



{{{x = (1 +- sqrt( 289 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (1 +- 17)/(2)}}} Take the square root of {{{289}}} to get {{{17}}}. 



{{{x = (1 + 17)/(2)}}} or {{{x = (1 - 17)/(2)}}} Break up the expression. 



{{{x = (18)/(2)}}} or {{{x =  (-16)/(2)}}} Combine like terms. 



{{{x = 9}}} or {{{x = -8}}} Simplify. 



So the answers are {{{x = 9}}} or {{{x = -8}}}