An introduction to the modern theory of equations, by Florian Cajori.

CHAPTER VIII

ELIMINATION
72. Resultants or Eliminants. Let us determine the condition that the two equations
f(x) aox2+ a x + a2 = 0,
F(x) = CoX2 + C1 x + C2 = 0,
shall have a root in common. Designate the roots of the
second equation by /,, /2. The necessary and sufficient condition that /p or /2 shall satisfy the equation f(x)= 0 is that
f(B1) or f(/2) shall vanish; in other words, that the product
f(/') f(/ () shall be zero. Multiplying together
f(l,) 0 0312 + C 1p, + a2,
f(92) - o022 + (Cl 18 + a2
we get
ao21,p22 + a,0a(/3,/22 + 12/3,) + ao02 (/132 + /22) + a2fil,832
+ t,,2(/1 + 92) + aC2.
Multiplying by c02 and substituting for the symmetric functions of /, and f, their values in terms of the coefficients of
F(x) = 0, we have
ao2C2 - aaC, a - 2 aoaaCo,c + a,2coc2 - alcCoC + a22cO'.
This expression is called the eliminant or resultant. Its vanishing is the condition that the given equations shall have a
root in common.
If from n equations involving n - 1 variables we eliminate
the variables and obtain an equation R = 0 involving only the
92