Question 352819
I'll do the first one to get you started


# 1


"One number is 2 more than 5 times another" means that {{{y=5x+2}}} and "Their product is 24" tells us that {{{xy=24}}} (x times y = 24)



{{{xy=24}}} Start with the second equation.



{{{x(5x+2)=24}}} Plug in {{{y=5x+2}}}



{{{5x^2+2x=24}}} Distribute.



{{{5x^2+2x-24=0}}} Subtract 24 from both sides.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=5}}}, {{{b=2}}}, and {{{c=-24}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(2) +- sqrt( (2)^2-4(5)(-24) ))/(2(5))}}} Plug in  {{{a=5}}}, {{{b=2}}}, and {{{c=-24}}}



{{{x = (-2 +- sqrt( 4-4(5)(-24) ))/(2(5))}}} Square {{{2}}} to get {{{4}}}. 



{{{x = (-2 +- sqrt( 4--480 ))/(2(5))}}} Multiply {{{4(5)(-24)}}} to get {{{-480}}}



{{{x = (-2 +- sqrt( 4+480 ))/(2(5))}}} Rewrite {{{sqrt(4--480)}}} as {{{sqrt(4+480)}}}



{{{x = (-2 +- sqrt( 484 ))/(2(5))}}} Add {{{4}}} to {{{480}}} to get {{{484}}}



{{{x = (-2 +- sqrt( 484 ))/(10)}}} Multiply {{{2}}} and {{{5}}} to get {{{10}}}. 



{{{x = (-2 +- 22)/(10)}}} Take the square root of {{{484}}} to get {{{22}}}. 



{{{x = (-2 + 22)/(10)}}} or {{{x = (-2 - 22)/(10)}}} Break up the expression. 



{{{x = (20)/(10)}}} or {{{x =  (-24)/(10)}}} Combine like terms. 



{{{x = 2}}} or {{{x = -12/5}}} Simplify. 



So the possible answers are {{{x = 2}}} or {{{x = -12/5}}} 

  
  
If you want integer only answers, then the only solution is {{{x=2}}}



So one number is 2 and the other is {{{y=5(2)+2=10+2=12}}}



So the two numbers are 2 and 12.