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Question Number 147500 by mathdanisur last updated on 21/Jul/21

x^2  - y^2  + 2x = 22  what is the number of complete  solutions that satisf the equation  (x;y).?

$${x}^{\mathrm{2}} \:-\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:=\:\mathrm{22} \\ $$$${what}\:{is}\:{the}\:{number}\:{of}\:{complete} \\ $$$${solutions}\:{that}\:{satisf}\:{the}\:{equation} \\ $$$$\left({x};{y}\right).? \\ $$

Commented by Mrsof last updated on 21/Jul/21

x^2 +2x+1−y^2 =22+1⇒(x+1)^2 −y^2 =23    it is four solution

$${x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}−{y}^{\mathrm{2}} =\mathrm{22}+\mathrm{1}\Rightarrow\left({x}+\mathrm{1}\right)^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{23} \\ $$$$ \\ $$$${it}\:{is}\:{four}\:{solution} \\ $$

Answered by Rasheed.Sindhi last updated on 21/Jul/21

I assumed ′number of integer solutions′  x^2  - y^2  + 2x = 22  x^2 +2x+1 - y^2   = 22+1  (x+1)^2 −y^2 =23  (x−y+1)(x+y+1)=23  23=1×23=23×1=−1×−23=−23×−1  ^• x−y+1=1 ∧ x+y+1=23  ^• x−y+1=−1 ∧ x+y+1=−23  ^• x−y+1=23 ∧ x+y+1=1  ^• x−y+1=−23 ∧ x+y+1=−1  Four non-singular  systems  of    simultaneous linear  equations.  ∴ FOUR integral solutions

$${I}\:{assumed}\:'{number}\:{of}\:\boldsymbol{{integer}}\:\boldsymbol{{solutions}}' \\ $$$${x}^{\mathrm{2}} \:-\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:=\:\mathrm{22} \\ $$$${x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}\:-\:{y}^{\mathrm{2}} \:\:=\:\mathrm{22}+\mathrm{1} \\ $$$$\left({x}+\mathrm{1}\right)^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{23} \\ $$$$\left({x}−{y}+\mathrm{1}\right)\left({x}+{y}+\mathrm{1}\right)=\mathrm{23} \\ $$$$\mathrm{23}=\mathrm{1}×\mathrm{23}=\mathrm{23}×\mathrm{1}=−\mathrm{1}×−\mathrm{23}=−\mathrm{23}×−\mathrm{1} \\ $$$$\:^{\bullet} {x}−{y}+\mathrm{1}=\mathrm{1}\:\wedge\:{x}+{y}+\mathrm{1}=\mathrm{23} \\ $$$$\:^{\bullet} {x}−{y}+\mathrm{1}=−\mathrm{1}\:\wedge\:{x}+{y}+\mathrm{1}=−\mathrm{23} \\ $$$$\:^{\bullet} {x}−{y}+\mathrm{1}=\mathrm{23}\:\wedge\:{x}+{y}+\mathrm{1}=\mathrm{1} \\ $$$$\:^{\bullet} {x}−{y}+\mathrm{1}=−\mathrm{23}\:\wedge\:{x}+{y}+\mathrm{1}=−\mathrm{1} \\ $$$$\mathrm{Four}\:\mathrm{non}-\mathrm{singular}\:\:\mathrm{systems}\:\:\mathrm{of}\:\: \\ $$$$\mathrm{simultaneous}\:\mathrm{linear}\:\:\mathrm{equations}. \\ $$$$\therefore\:\mathrm{FOUR}\:\mathrm{integral}\:\mathrm{solutions} \\ $$

Commented by mathdanisur last updated on 21/Jul/21

thank you Sir

$${thank}\:{you}\:{Sir} \\ $$

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