the difference of two positive numbers is 4. The sum of their squares is 170. What are the numbers?

Question

 I have the answers 7 and 11 by doing guess and check, but I want to know another way of working this problem. 

Answer 1

Note:    ONE SOLUTION  because "two positive numbers"

Answer 2

 

 X-Y = 4 ---> X= y+4

X^2 + Y^2 = 170

 

(Y+4)^2 + y^2 = 170

 

Y^2 + 8y + 16 + y^2 = 170

 

2y^2 + 8y + 16 = 170

 

2y^2 + 8y - 154 = 0

 

y^2 + 4y - 77 = 0

 

( y    +   11)( y  -    7 ) = 0

 

y = -11 or y=7

 

 

y=-11 ---> x - (-11) = 4

                 x+11= 4

                x= -7

 

x=-7 and y=-11 indeed differ by 4 and their squares total

49 + 121 = 170

 

Likewise for y = 7, 

x - 7 = 4 ---> x = 11

So x=7 and y = 11 indeed differ by 4 and their squares

total 49 + 121 = 170. This is the POSITIVE version of

the previous answer.

 

The two solutions are:

{-7 and -11}  and {7 and 11}