For how many integer values of a does the equation x^2 + ax + 28a = 0 have integer solutions for x?
+26398 For how many integer values of a does the equation
x2+ax+28a=0
have integer solutions for x?
Let the roots are r1 and r2:
Vieta:
x2+a⏟=−(r1+r2)x+28a⏟=r1r2a=−(r1+r2)28a=r1r228(−(r1+r2))=r1r2−28r1−28r2=r1r2r1r2+28r1+28r2=0(r1+28)(r2+28)−28∗28=0(r1+28)(r2+28)=28∗28(r1+28)(r2+28)=784
The divisors of 784 are:
1 | 2 | 4 | 7 | 8 | 14 | 16 | 28 |
49 | 56 | 98 | 112 | 196 | 392 | 784 (15 divisors)
(r1+28)(r2+28)=1∗784=2∗392=4∗196=7∗112=8∗98=14∗56=16∗49=28∗28
(r1+28)r1(r2+28)r2a=−r1−r211−28=−27784784−28=75627−756=−72922−28=−26392392−28=36426−364=−33844−28=−24196196−28=16824−168=−14477−28=−21112112−28=7421−74=−5388−28=−209898−28=7020−70=−501414−28=−145656−28=2814−28=−141616−28=−124949−28=2112−21=−92828−28=02828−28=0−0−0=04949−28=211616−28=−12−21+12=−95656−28=281414−28=−14−28+14=−149898−28=7088−28=−20−70+20=−50112112−28=7477−28=−21−74+21=−53196196−28=16844−28=−24−168+24=−144392392−28=36422−28=−26−364+26=−338784784−28=75611−28=−27−756+27=−729
Distinct Integer values of a are 7 without a=0: {−9, −14, −50, −53 −144, −338, −729}
