Question 4403
Let x = first number
y = second number


Write two equations based upon the first two sentences.
x-y = 7
xy = 1


Solve for x in the first equation by adding +y to each side.
x-y = 7
x-y+y= 7 + y
x = 7+y


Substitute x = 7+y into the second equation for x:

xy = 1
(7+y)y = 1


Next use the distributive property, and since a quadratic equation results, set it equal to zero.
{{{7y + y^2 = 1}}}

{{{Y^2 + 7y - 1= 0}}}


This obviously does not factor, which may be why you had a problem with it.  There are no whole number solutions that work!! Use the quadratic formula to solve it:
{{{y = (-b+-sqrt(b^2 - 4ac))/(2a)}}}, where a= 1, b=7, c=-1

{{{y = (-7+-sqrt(7^2 - 4*1*(-1)))/(2*1)}}}

{{{y = (-7+-sqrt(47 + 4))/2}}}

{{{y = (-7+-sqrt(53))/2}}}


It appears that there are two solutions

{{{y = (-7+sqrt(53))/2}}} and   {{{y = (-7-sqrt(53))/2}}}.
However, according to the problem, these must be POSITIVE numbers, which rules out the second above.

The solution is {{{y = (-7+sqrt(53))/2}}}  
and {{{x = 7 + y}}}
    or {{{x = 7 + (-7+sqrt(53))/2}}}  
 or {{{x= (14 - 7 + sqrt (53))/2}}} = {{{(7+ sqrt (53))/2}}}


To check, show that their difference is 7 and their product is 1.  (I sure don't want to post this so everyone can see, until I make sure that it works!!)


Difference = {{{(7+ sqrt (53))/2}}} - {{{(-7+ sqrt (53))/2}}}
Difference = {{{(7 + sqrt (53) -(-7) - sqrt (53)) /2}}} = {{{14/2}}} ={{{7}}}


Product =  {{{(7+ sqrt (53))/2 * (-7+ sqrt (53))/2}}} FOIL this out!!
      
{{{  (-49 +7 *sqrt (53) - 7 *sqrt (53) + 53)/ (2*2)}}}

{{{4/4}}} = {{{1}}}


Final Answer:  {{{(7+ sqrt (53))/2}}},  {{{(-7+ sqrt (53))/2}}}
  


R^2 at SCC