Question 866162
Let a = the smaller positive integer and
let b = the larger positive integer
Then we have two equations
(1) b - a = 5 and
(2) a + b^2 = 301
Solve (1) for b and substitute it into (2) and get
(3) a + (5 + a)^2 = 301 or
(4) a + 25 + 10a +a^2 - 301 = 0 or
(5) a^2 + 11a - 276 = 0
Use the quadratic equation solution formul
(6) {{{(-11+-sqrt(11^2-4*1*(-276))/(2*1))}}} and get
(7) a = {-23,12}
We select a = 12, because it must be a positive integer. Using (1) we get
(8) b = 5 + 12 or
(9) b = 17
Check these values using (2).
Is (12 + 17^2 = 301)?
Is (12 + 289 = 301)?
Is (301 = 301)? Yes
Answer: The two positive integers are 12 and 17.
PS a = -23 and b = -18 should also be a solution. Try it in (2).