The sum of two numbers is 25 and their differences is 13 find their product?

Let a and b be the two numbers.

a + b = 25 ...... [1]
a - b = 13 ...... [2]


Method 1 :

[1] + [2] :
2a = 38
a = 19

[1] - [2] :
2b = 12
b = 6

The product of the two numbers
= a × b
= 19 × 6
= 114


Method 2
[1]² : a² + 2ab + b² = 625 ...... [3]
[2]² : a² - 2ab + b² = 169 ...... [4]

[3] - [4] :
4ab = 456
ab = 114

The product of the two numbers = 114

x + y = 25
x - y = 13______ADD

2x = 38
x = 19
y= 6

Product is 114

19 x 6 = 114.

A + B = 25
and
A - B = 13
so, combining the two together you get
(A+A) + (B-B) = (25+13)
which is
2A = 38

A therefore = 19
and B = (25-19) .. 6

and finally .. 19 x 6 = 114

a+b=25..[1]. a-b=13..[2].{[1]+[2]}-->2a = 38, ie., a = 19. Then [1]--> b = 6 & ab = 114.

x+y=a
x-y=b

(x+y)^2 = a^2 = x^2 + 2xy + y^2
(x-y)^2 = b^2 = x^2 - 2xy + y^2

a^2 - b^2 = 4xy
xy = [a^2 - b^2] / 4

or

2x = a+b
x = (a+b)/2

2y = a-b
y = (a-b)/2

xy = (a+b)(a-b) / 4
xy = [a^2 - b^2] / 4

sum = 25
difference = 13

product = (25^2 - 13^2)/4
= (625 - 169) / 4 = 114

(25+13)/2 * (25-13)/2
= 38/2 * 12/2 = 19 * 6 = 114

x+y=25
x-y=13
2x=38; x=19, y=6
19*6=114