Question

If sum of two numbers and sum of squares of the numbers are given, find their difference

Answer 1

Δ3


2222
As usual, the first n in the table is zero, which isn't a natural number. 
Because Δ3 is a constant, the sum is a cubic of the form 
an3+bn2+cn+d, [1.0]

and we can find the coefficients using simultaneous equations, which we can make as we wish, as we know how to add squares to the table and to sum them, even if we don't know the formula. 

In the table below, we create three equations, noting that d=0 from the first one (revealing the reason for the non-natural number zero)

n0123Sn01514
d=0a+b+c=18a+4b+2c=527a+9b+3c=14
Rewriting our equations:
a+b+c=1 [1.1]
8a+4b+2c=5 [1.2]
27a+9b+3c=14 [1.3]

Using 1.1 with 1.2 and 1.3 we can make two new equations:
6a+2b=3 [1.4]
24a+6b=11 [1.5]

By subtracting 3x Equation 1.4 from 1.5, we get:
6a=2
So a=1/3 [Also noting, by the way, that Δ3/3!=1/3]

Answer 2

Let no.s be x and y
given:-x+y=9 and x^2 -y^2=9
then...x^2-y^2=(x+y) (x-y)
--put values..
9=(9) (x-y)
9/9=(x-y)
1=x-y
now x +y =9
x -y = 1
---------------
2x =10
--------------
x=10/2= 5
as x-y= 1
therefore...5-y=1
4=y