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Quadratic Invariant


Given the binary quadratic form

 ax^2+2bxy+cy^2
(1)

with polynomial discriminant b^2-ac, let

x=pX+qY
(2)
y=rX+sY.
(3)

Then

 a(pX+qY)^2+2b(pX+qY)(rX+sY)+c(rX+sY)^2 
 =AX^2+2BXY+CY^2,
(4)

where

A=ap^2+2bpr+cr^2
(5)
B=apq+b(ps+qr)+crs
(6)
C=aq^2+2bqs+cs^2,
(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).
(8)

Surprisingly, this is the same discriminant as before, but multiplied by the factor (ps-rq)^2. The quantity ps-rq is called the quadratic invariant modulus.


See also

Algebraic Invariant, Quadratic

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Cite this as:

Weisstein, Eric W. "Quadratic Invariant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/QuadraticInvariant.html

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