Question 195570
{{{2/x-x/8=3/4}}} Start with the given equation.



{{{8*cross(x)(2/cross(x))-cross(8)x(x/cross(8))=cross(8)^2*x(3/cross(4))}}} Multiply <font size="4"><b>every</b></font> term on both sides by the LCD {{{8x}}}. Doing this will eliminate all of the fractions.



{{{8(2)-x(x)=2x(3)}}} Cancel out and simplify



{{{16-x^2=6x}}} Multiply



{{{16-x^2-6x=0}}} Get all terms to the left side.



{{{-x^2-6x+16=0}}} Rearrange the terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=-1}}}, {{{b=-6}}}, and {{{c=16}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-6) +- sqrt( (-6)^2-4(-1)(16) ))/(2(-1))}}} Plug in  {{{a=-1}}}, {{{b=-6}}}, and {{{c=16}}}



{{{x = (6 +- sqrt( (-6)^2-4(-1)(16) ))/(2(-1))}}} Negate {{{-6}}} to get {{{6}}}. 



{{{x = (6 +- sqrt( 36-4(-1)(16) ))/(2(-1))}}} Square {{{-6}}} to get {{{36}}}. 



{{{x = (6 +- sqrt( 36--64 ))/(2(-1))}}} Multiply {{{4(-1)(16)}}} to get {{{-64}}}



{{{x = (6 +- sqrt( 36+64 ))/(2(-1))}}} Rewrite {{{sqrt(36--64)}}} as {{{sqrt(36+64)}}}



{{{x = (6 +- sqrt( 100 ))/(2(-1))}}} Add {{{36}}} to {{{64}}} to get {{{100}}}



{{{x = (6 +- sqrt( 100 ))/(-2)}}} Multiply {{{2}}} and {{{-1}}} to get {{{-2}}}. 



{{{x = (6 +- 10)/(-2)}}} Take the square root of {{{100}}} to get {{{10}}}. 



{{{x = (6 + 10)/(-2)}}} or {{{x = (6 - 10)/(-2)}}} Break up the expression. 



{{{x = (16)/(-2)}}} or {{{x =  (-4)/(-2)}}} Combine like terms. 



{{{x = -8}}} or {{{x = 2}}} Simplify. 



So the answers are {{{x = -8}}} or {{{x = 2}}}