Question 1202304
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Answer: The two numbers are 13 and 7



Work Shown:


x and y are the two numbers
Let x > y
I'll assume both are integers.


x^2 + y^2 = 218 ... sum of squares
y^2 = 218 - x^2
y = sqrt(218 - x^2)


x-y = difference
y(x-y) = 42
xy - y^2 = 42
x*sqrt(218-x^2) - (218 - x^2) = 42
x*sqrt(218-x^2) - 218 + x^2 = 42
x*sqrt(218-x^2) - 218 + x^2 - 42 = 0


Use a graphing calculator such as Desmos or GeoGebra to plot
f(x) = x*sqrt(218-x^2) - 218 + x^2 - 42

 
You should find one of the x intercepts is x = 13
y = sqrt(218 - x^2)
y = sqrt(218 - 13^2)
y = 7


The two numbers are 13 and 7.


Check:
7^2 + 13^2 = 49 + 169 = 218
7*(13-7) = 7*6 = 42
The answers are confirmed.
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