First equation roots:

x = (A +/- sqrt(A^2 - 4*B) / 2

Second equation roots:

x = (B +/- sqrt(B^2 - 4*A) / 2

4*B <= A^2

4*A <= B^2

4*sqrt(A) <= A^2, remembering A is positive

4*sqrt(A) <= (sqrt(A))^4

(sqrt(A))^3 >= 4

A >= 2.5198... and since integer, >= 3

Same for B

sqrt(A^2 - 4*B) must be of same parity as A, and must be smaller than A so as to make both roots positive. (Each equation must have positive roots -- singular equation, plural roots).

The same with sqrt(B^2 - 4*A) vs B.

Found that satisfy:

A B     roots

4 4    2 2

5 6    4.5 4.5

Also B and A can be interchanged due to the symmetry of the equations. The roots would be reversed as well, but the roots are the same in each case.

However, 4.5 is not an integer so A=4; B=4 would seem the only pair.

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 For tot = 6 To 10000

   For a = 3 To tot / 2

     b = tot - a

     If 4 * b <= a * a Then

      If 4 * a <= b * b Then

        tst = Sqr(a * a - 4 * b)

        If tst = Int(tst) And (tst - a) Mod 2 = 0 And tst < a Then

          tst = Sqr(b * b - 4 * a)

          If tst = Int(tst) And (tst - b) Mod 2 = 0 And tst < b Then

            

            Text1.Text = Text1.Text & a & Str(b) & "    "

            Text1.Text = Text1.Text & (a + tst) / 2 & Str((a + tst) / 2) & crlf

            

          End If

        End If

      End If

     End If

     DoEvents

   Next a

 Next tot

 

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub