Question 169972
"One number is 6 less than another." translates to {{{y=x-6}}}



"The product of the number is 72" translates to {{{x*y=72}}}



{{{x*y=72}}} Start with the second equation



{{{x(x-6)=72}}} Plug in {{{y=x-6}}}



{{{x^2-6x=72}}} Distribute.



{{{x^2-6x-72=0}}} Subtract 72 from both sides.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-6}}}, 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 = (-(-6) +- sqrt( (-6)^2-4(1)(-72) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-6}}}, and {{{c=-72}}}



{{{x = (6 +- sqrt( (-6)^2-4(1)(-72) ))/(2(1))}}} Negate {{{-6}}} to get {{{6}}}. 



{{{x = (6 +- sqrt( 36-4(1)(-72) ))/(2(1))}}} Square {{{-6}}} to get {{{36}}}. 



{{{x = (6 +- sqrt( 36--288 ))/(2(1))}}} Multiply {{{4(1)(-72)}}} to get {{{-288}}}



{{{x = (6 +- sqrt( 36+288 ))/(2(1))}}} Rewrite {{{sqrt(36--288)}}} as {{{sqrt(36+288)}}}



{{{x = (6 +- sqrt( 324 ))/(2(1))}}} Add {{{36}}} to {{{288}}} to get {{{324}}}



{{{x = (6 +- sqrt( 324 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (6 +- 18)/(2)}}} Take the square root of {{{324}}} to get {{{18}}}. 



{{{x = (6 + 18)/(2)}}} or {{{x = (6 - 18)/(2)}}} Break up the expression. 



{{{x = (24)/(2)}}} or {{{x =  (-12)/(2)}}} Combine like terms. 



{{{x = 12}}} or {{{x = -6}}} Simplify. 



So the answers are {{{x = 12}}} or {{{x = -6}}} 



This means that the numbers are:



12 and 6


OR...


-6 and -12