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Question Number 199732 by Rupesh123 last updated on 08/Nov/23

Answered by Rasheed.Sindhi last updated on 08/Nov/23

y=((x^2 +2)/(x+3))=((x^2 −9+9+2)/(x+3)) =(((x−3)(x+3)+11)/(x+3)) =(x−3)+((11)/(x+3))∈Z⇒(x+3)∣11 x+3=±1,±11 x=−2,−4,8,−14 y=((x^2 +2)/(x+3)) y=(((−2)^2 +2)/((−2)+3)),(((−4)^2 +2)/((−4)+3)),(((8)^2 +2)/((8)+3)),(((−14)^2 +2)/((−14)+3)) =6,−18,6,−18 (x,y)=(−2,6),(−4,−18),(8,6),(−14,−18)

y=x2+2x+3=x2−9+9+2x+3=(x−3)(x+3)+11x+3=(x−3)+11x+3∈Z⇒(x+3)∣11x+3=±1,±11x=−2,−4,8,−14y=x2+2x+3y=(−2)2+2(−2)+3,(−4)2+2(−4)+3,(8)2+2(8)+3,(−14)2+2(−14)+3=6,−18,6,−18(x,y)=(−2,6),(−4,−18),(8,6),(−14,−18)

Commented by Rupesh123 last updated on 08/Nov/23

Perfect, sir!

Answered by Rasheed.Sindhi last updated on 08/Nov/23

AnOther Way y(x+3)−x^2 =2 y(x+3)−x^2 +9=2+9 y(x+3)−(x^2 −9)=11 y(x+3)−(x−3)(x+3)=11 (x+3)(y−x+3)=11=1×11=−1×−11 { ((x+3=1 ∧ y−x+3=11)),((x+3=11 ∧ y−x+3=1)),((x+3=−1 ∧ y−x+3=−11)),((x+3=−11 ∧ y−x+3=−1)) :} { ((x=−2 ∧ y=x+8=−2+8=6)),((x=8 ∧ y=x−2=8−2=6)),((x=−4 ∧ y=x−14=−4−14=−18)),((x=−14 ∧ y=x−4=−14−4=−18)) :} (x,y)=(−2,6),(8,6),(−4,−18),(−14,−18)

AnOtherWayy(x+3)−x2=2y(x+3)−x2+9=2+9y(x+3)−(x2−9)=11y(x+3)−(x−3)(x+3)=11(x+3)(y−x+3)=11=1×11=−1×−11{x+3=1∧y−x+3=11x+3=11∧y−x+3=1x+3=−1∧y−x+3=−11x+3=−11∧y−x+3=−1{x=−2∧y=x+8=−2+8=6x=8∧y=x−2=8−2=6x=−4∧y=x−14=−4−14=−18x=−14∧y=x−4=−14−4=−18(x,y)=(−2,6),(8,6),(−4,−18),(−14,−18)

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