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remark-to-question-195301-and-similar-ones-x-2-y-a-x-y-2-b-a-b-gt-0-how-many-solutions-depending-on-a-b-

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Question Number 195356 by MJS_new last updated on 31/Jul/23
remark to question 195301 and similar ones x^2 +y=a x+y^2 =b a, b >0 how many solutions depending on a, b?
remarktoquestion195301andsimilaronesx2+y=ax+y2=ba,b>0howmanysolutionsdependingona,b?
Answered by MJS_new last updated on 31/Jul/23
y=a−x^2 inserting in the 2^(nd) equation gives x^4 −2ax+x+a^2 −b=0 possible cases: I. 4 distinct real solutions a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))>0 II. 2 distinct and 1 double real solutions b≠(4/3)a^2 ∧a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))=0 III. 1 single and 1 triple real solutions b=(4/3)a^2 ∧a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))=0 ⇔ a=b=(3/4) IV. 2 distinct real and 1 pair of conjugated complex solutions a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))<0
y=a−x2insertinginthe2ndequationgivesx4−2ax+x+a2−b=0possiblecases:I.4distinctrealsolutionsa3−a2b2−98ab+b3+27256>0II.2distinctand1doublerealsolutionsb≠43a2∧a3−a2b2−98ab+b3+27256=0III.1singleand1triplerealsolutionsb=43a2∧a3−a2b2−98ab+b3+27256=0⇔a=b=34IV.2distinctrealand1pairofconjugatedcomplexsolutionsa3−a2b2−98ab+b3+27256<0
Commented by MJS_new last updated on 31/Jul/23
with a=11∧b=7 we get case I
witha=11∧b=7wegetcaseI
Commented by Frix last updated on 31/Jul/23
And without a,b >0?
Andwithouta,b>0?
Answered by MJS_new last updated on 31/Jul/23
without the restriction a, b >0: a, b ∈R I. 4 distinct real solutions a>0∧b>0∧a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))<0 II. 2 distinct and 1 double real solutions a>0∧b>0∧b≠(4/3)a^2 ∧a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))=0 III. 1 single and 1 triple real solutions a=b=(3/4) IV. 2 distinct real and 1 pair of conjugated complex solutions a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))>0 V. 1 double real and 1 pair of conjugated complex solutions a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))=0∧(a<0∨b<0) VI. 2 pairs of conjugated complex solutions a^3 −a^2 b^2 −(9/8)ab+b^3 +((27)/(256))<0∧(a<0∨b<0)
withouttherestrictiona,b>0:a,b∈RI.4distinctrealsolutionsa>0∧b>0∧a3−a2b2−98ab+b3+27256<0II.2distinctand1doublerealsolutionsa>0∧b>0∧b≠43a2∧a3−a2b2−98ab+b3+27256=0III.1singleand1triplerealsolutionsa=b=34IV.2distinctrealand1pairofconjugatedcomplexsolutionsa3−a2b2−98ab+b3+27256>0V.1doublerealand1pairofconjugatedcomplexsolutionsa3−a2b2−98ab+b3+27256=0∧(a<0∨b<0)VI.2pairsofconjugatedcomplexsolutionsa3−a2b2−98ab+b3+27256<0∧(a<0∨b<0)

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