The quadratic invariant of the binary quadratic form
 |
(1)
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is its polynomial discriminant 
. Under the change of variables
 |  |  |
(2)
|
 |  |  |
(3)
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the transformed form is
 |
(4)
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where
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
so
![B^2-AC=[a^2p^2q^2+b^2(ps+qr)^2+c^2r^2s^2
+2abpq(ps+qr)+2acpqrs+2bcrs(ps+qr)]
-(ap^2+2bpr+cr^2)(aq^2+2bqs+cs^2)
=a^2p^2q^2+b^2p^2s^2+2b^2pqrs+b^2q^2r^2+c^2r^2s^2
+2abp^2qs+2abpq^2r+2acpqrs+2bcprs^2+2bcqr^2s
-a^2p^2q^2-2abp^2qs-acp^2s^2-2abpq^2r-4b^2pqrs
-2bcprs^2-acq^2r^2-2bcqr^2s-c^2r^2s^2
=b^2p^2s^2-2b^2pqrs+b^2q^2r^2+2acpqrs-acp^2s^2
-acq^2r^2
=p^2s^2(b^2-ac)+q^2r^2(b^2-ac)-2pqrs(b^2-ac)
=(b^2-ac)(p^2s^2-2pqrs+q^2r^2)
=(ps-rq)^2(b^2-ac).](/images/equations/QuadraticInvariant/NumberedEquation3.svg) |
(8)
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Surprisingly, this is the same discriminant as before, but multiplied by the factor 
.
The quantity 
is called the quadratic invariant modulus.
