Let's say those numbers are X and Y
We know that their sum is 30:
X + Y = 30
Since their squares are X² and Y², then the sum of their squares is
X² + Y² = 468
Now, we have two equations:
X + Y = 30
X² + Y² = 468
Finding Y in the first equation, so we can replace it in the second:
Y = 30 - X
then,
X² + (30 - X)² = 468
since
(30 - X)² = 30² - 2 (30X) + X²
= 900 - 60X + X²
Then
X² + (30 - X)² = 468
X² + 900 - 60X + X² = 468
2X² - 60X +900 - 468 = 0
2X² - 60X + 432 = 0
Dividing both sides by 2
2X² - 60X + 432 = 0
X² - 30X + 216 = 0
We complete the perfect square polynomial
X² - 30X + 216 = 0
X² - 2 (15X) = -216
X² - 2 (15X) + 15² = -216 + 15²
Solving the perfect square polynomial:
X² - 2 (15X) + 15² = -216 + 15²
(X - 15)² = -216 + 15² = 225 - 216 = 9
(X - 15)² = 9
√9 = 3
X - 15 = ±3
X = ±3 +15
We have two possibilities for X:
X₁ = 3 + 15
X₂ = -3 + 15
Then
X₁ = 18
X₂ = 12
Replacing them in the first equation:
X₁ + Y₁ = 30
18 + Y₁ = 30
Y₁ = 30 - 18
Y₁ = 12
X₂ + Y₂ = 30
12 + Y₂ = 30
Y₂ = 30 - 12
Y₂ = 18
Then if X = 12, then Y = 18
if X = 18 then Y = 12
Then their product will be always:
18 · 12 = 216
