Question

20 point

The difference of two positive integers is 5 and the sum of their squares is 433. Find the integers.

Answer 1

The two positive integers are 17 and 12.

Explanation:

Let the two positive integers be a and b.

The information provided is:

`a-b=5\\a^(2)+ b^(2) = 433`

The square of the difference between two positive integers is:

`(a-b)^(2) = a^(2)+b^(2) -2ab`

Use the provided information to determine the value of 2ab as follows:

`(a-b)^(2) = a^(2)+b^(2) -2ab\\5^(2) = 433 - 2ab\\2ab=408`

The square of the sum of two positive integers is:

`(a+b)^(2) = a^(2)+b^(2) +2ab\\=433+408\\=841\\(a+b)=√(841)\\ =29`

Now,

`a+b=29...(i)\\a-b=5...(ii)`

Add (i) and (ii) and solve:

`2a=34\\a=(34)/(2) \\=17`

Substitute a = 17 in (i) to compute b:

`a+b=29\\17+b=29\\b=12`

Thus, the two positive integers are 17 and 12.