Question

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

Answer 1

Answer:

The two positive integers are 17 and 12.

Step-by-step 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:

$(a+b)^{2} = a^{2}+b^{2} +2ab\\=433+408\\=841\\(a+b)=\sqrt{841}\\ =29$

Now,

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

Add (i) and (ii) and solve:

$2a=34\\a=\frac{34}{2} \\=17$

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.