How to solve tasks of this type from the exam in computer science?

Question

Programming

What is the smallest base of the positional number system x ,
in which 225 _x \u003d 405 _y , x and y are the bases of the number systems?
Write your answer as an integer.
I met only tasks where the basis of only one number system was the unknown. I tried to enter a function y(x)and do it through the derivative - it did not work. The correct answer for this problem is: x = 8.

Answer 1

225 _x = 2∙x ² + 2∙x + 5 = 4∙y ² + 5 = 405 _y
2∙x ² + 2∙x = 4∙y ²
y _min = max(4, 0, 5) + 1 = 6
2∙x ² + 2∙x = 4∙6 ² = 144
x ² + x - 72 = 0
D = 1 + 4*72 = 289
x ₁ = -(1 + 17)/2 = -9
x ₂ = -(1 - 17)/2 = 8
PS It is also possible in a more general form:
2∙x ² + 2∙x = 4∙y ²
x ² + x - 2∙y ² = 0
D = 1 + 8∙y ²
x ₁= -(1 + sqrt(1 + 8∙y ² ))/2
x ₂ = -(1 - sqrt(1 + 8∙y ² ))/2

Answer 2

I hope I wrote everything clearly enough.