equal Riots of an Equation by Divifon. 
419 
theorem II. 
J 
If the biquadratic equation x 4 — px +qx z — rx -\-s ~o has two 
) — f> r — 16s p _ 
4 A_ *-3 P' 
equal roots , make A * > B = £nX» C “ IX+T»» ami D ~ 
and you will have x — 
3 pp-%1 
D-B. 
4 A+3/» 7 ^ A ~ c 
A fynthetical demonftration of this theorem would be very 
long : tlie investigation is as follows. 
It has been demon ft rated by the writers on algebra, that, it 
a biquadratic equation, as x 4 — px* +qx* rx +s — o, has two 
equal roots, one of them may be had from the equation 
4* 3 _ 2 px ' 1 2 qx - r = o. Multiply this equation by a, and the 
original one by 4, and take the difference of the two, which will 
be px * — iqx ' 1 4- qr.v — 45 — o. Again, if this equation be multi- 
plied by 4, and the other cubic by />, and their difference taken, 
we {hall have $pp — 8^ x a' + iir— ipq x x +pr — — o, 01 
v * + I2r ~ 2 ^ x 4^ - o, or x* 4- Ax + B = o, putting A and 
3PP- 8 ? 3 pp-%1 
B for the known quantities in the iecond and third terms. Now 
multiply this equation by 4A, and take the firft cubic from it, 
and we fhall have 4A 4- $p x a 2 + 4B — 2 q x x + r — c, which 
being divided by 4A + $p, and C and D put eqaal to 
4B-2y an j — I — . refpe&ively, gives x* 4- Cv -p D = o ; and 
4 A+ 3 B 4 A +3B p .. 
this equation being taken from the other quadratic, there remains 
A^C xa+B-D-o; confequently x = * ^E. L 
COROLLARY i. From tbs above inveftigation it appears, that 
one of the equal roots may alio be obtained from either of thele 
two quadratic equations, of which the firft fee ms moft eligible,. 
I i i 2 
