Question 221402
"The difference of two positive numbers is 5" means that {{{x-y=5}}} (note: this means that {{{x>y}}})


Also, because "the sum of their squares is 157" we know that {{{x^2+y^2=157}}}



{{{x-y=5}}} Start with the first equation.



{{{-y=5-x}}} Subtract x from both sides.



{{{y=x-5}}} Multiply both sides by -1 and rearrange the terms.



{{{x^2+y^2=157}}} Move onto the second equation.



{{{x^2+(x-5)^2=157}}} Plug in {{{y=x-5}}} 



{{{x^2+x^2-10x+25=157}}} FOIL



{{{x^2+x^2-10x+25-157=0}}} Subtract 157 from both sides.



{{{2x^2-10x-132=0}}} Combine like terms.



Notice that the quadratic {{{2x^2-10x-132}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=2}}}, {{{B=-10}}}, and {{{C=-132}}}



Let's use the quadratic formula to solve for "x":



{{{x = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{x = (-(-10) +- sqrt( (-10)^2-4(2)(-132) ))/(2(2))}}} Plug in  {{{A=2}}}, {{{B=-10}}}, and {{{C=-132}}}



{{{x = (10 +- sqrt( (-10)^2-4(2)(-132) ))/(2(2))}}} Negate {{{-10}}} to get {{{10}}}. 



{{{x = (10 +- sqrt( 100-4(2)(-132) ))/(2(2))}}} Square {{{-10}}} to get {{{100}}}. 



{{{x = (10 +- sqrt( 100--1056 ))/(2(2))}}} Multiply {{{4(2)(-132)}}} to get {{{-1056}}}



{{{x = (10 +- sqrt( 100+1056 ))/(2(2))}}} Rewrite {{{sqrt(100--1056)}}} as {{{sqrt(100+1056)}}}



{{{x = (10 +- sqrt( 1156 ))/(2(2))}}} Add {{{100}}} to {{{1056}}} to get {{{1156}}}



{{{x = (10 +- sqrt( 1156 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (10 +- 34)/(4)}}} Take the square root of {{{1156}}} to get {{{34}}}. 



{{{x = (10 + 34)/(4)}}} or {{{x = (10 - 34)/(4)}}} Break up the expression. 



{{{x = (44)/(4)}}} or {{{x =  (-24)/(4)}}} Combine like terms. 



{{{x = 11}}} or {{{x = -6}}} Simplify. 



So the possible solutions are {{{x = 11}}} or {{{x = -6}}} 

  
  
However, it's stated that the numbers are positive. So the only answer is {{{x=11}}}



Now just plug that value into {{{x-y=5}}} and solve for 'y' to get {{{11-y=5}}} ---> {{{y=11-5=6}}}



So the two numbers are {{{x=11}}} and {{{y=6}}}



Notice how 11-6=5. So their difference is 5


Also, take note how {{{11^2+6^2=121+36=157}}}.


So this confirms our answer.