Answer:
[tex]17[/tex] and [tex]20[/tex].
Step-by-step explanation:
Let [tex]x[/tex] denote the larger one of the two positive integers ([tex]x > 0[/tex].) The smaller integer would be [tex](x - 3)[/tex]. The square of the larger integer would be [tex]x^{2}[/tex].
The sum of the smaller integer and the square of the larger integer is [tex](x^{2} + (x - 3))[/tex]. If this sum is equal to [tex]417[/tex], then:
[tex]x^{2} + x - 3 = 417[/tex].
[tex]x^{2} + x - 420 = 0[/tex].
Solve this quadratic equation for [tex]x[/tex] through factorization. Since [tex]420 = 21 \times 20[/tex]:
[tex]x^{2} + x - (21 \times 20) = 0[/tex].
[tex]x^{2} + 21\, x - 20\, x + (21)\, (-20) = 0[/tex].
[tex](x + 21)\, (x - 20) = 0[/tex].
Thus, either [tex]x = 20[/tex] or [tex]x = (-21)[/tex] by the Factor Theorem. However, since [tex]x > 0[/tex] (the two integers are both positive,) only [tex]x = 20[/tex] is a valid solution.
Hence, the larger one of the two integers would be [tex]20[/tex]. The smaller one of the two integers would be [tex](x - 3) = (20 - 3) = 17[/tex].
