Question

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

Answer 1

12 and 17

Explanation:

Let the two integers be m and n

`m - n = 5`

`m^2+n^2=433`

From the first equation,

`m=5+n`

Substitute this in the second equation.

`(5+n)^2+n^2=433`

`25+10n+n^2+n^2=433`

`25+10n+2n^2=433`

`2n^2+10n-408=0`

Divide both sides 2

`n^2+5n-204=0`

Factorise to get

`(n-12)(n+17)=0`

Therefore,
`n=12` or
`-17`

But
`n`is a positive integer. Therefore
`n=12`

From the first equation,

`m-n =5`

`m-12 =5=17`

The two integers are 12 and 17.

Let's check

17 - 12 = 5

`17^2+11^2 = 289 + 144 = 433`