Question

The difference two positive integer is 3. If the smaller is added to the square of the larger the sun is 417 find the integers

Answer 1

`17` and
`20`.

Explanation:

Let
`x` denote the larger one of the two positive integers (
`x > 0`.) The smaller integer would be
`(x - 3)`. The square of the larger integer would be
`x^(2)`.

The sum of the smaller integer and the square of the larger integer is
`(x^(2) + (x - 3))`. If this sum is equal to
`417`, then:

`x^(2) + x - 3 = 417`.

`x^(2) + x - 420 = 0`.

Solve this quadratic equation for
`x` through factorization. Since
`420 = 21 * 20`:

`x^(2) + x - (21 * 20) = 0`.

`x^(2) + 21\, x - 20\, x + (21)\, (-20) = 0`.

`(x + 21)\, (x - 20) = 0`.

Thus, either
`x = 20` or
`x = (-21)` by the Factor Theorem. However, since
`x > 0` (the two integers are both positive,) only
`x = 20` is a valid solution.

Hence, the larger one of the two integers would be
`20`. The smaller one of the two integers would be
`(x - 3) = (20 - 3) = 17`.