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Determine-all-pairs-of-real-numbers-a-b-such-that-a-6-b-4-b-6-a-4-4-




Question Number 146377 by mathdanisur last updated on 13/Jul/21
Determine all pairs of real numbers  (a;b) such that:  a^6 −b^4 =b^6 −a^4 =4
$${Determine}\:{all}\:{pairs}\:{of}\:{real}\:{numbers} \\ $$$$\left({a};{b}\right)\:{such}\:{that}: \\ $$$${a}^{\mathrm{6}} −{b}^{\mathrm{4}} ={b}^{\mathrm{6}} −{a}^{\mathrm{4}} =\mathrm{4} \\ $$
Commented by mr W last updated on 13/Jul/21
a=b=±(√2)
$${a}={b}=\pm\sqrt{\mathrm{2}} \\ $$
Commented by mathdanisur last updated on 13/Jul/21
thankyou Ser, solution
$${thankyou}\:{Ser},\:{solution} \\ $$
Commented by mr W last updated on 13/Jul/21
x=a^2   y=b^2   x^3 −y^2 =y^3 −x^2 =4  x^3 +x^2 =y^3 +y^2   ⇒x=y  x^3 −x^2 =4  x^2 (x−1)=4  ⇒x=2⇒a=±(√2)  ⇒y=2⇒b=±(√2)
$${x}={a}^{\mathrm{2}} \\ $$$${y}={b}^{\mathrm{2}} \\ $$$${x}^{\mathrm{3}} −{y}^{\mathrm{2}} ={y}^{\mathrm{3}} −{x}^{\mathrm{2}} =\mathrm{4} \\ $$$${x}^{\mathrm{3}} +{x}^{\mathrm{2}} ={y}^{\mathrm{3}} +{y}^{\mathrm{2}} \\ $$$$\Rightarrow{x}={y} \\ $$$${x}^{\mathrm{3}} −{x}^{\mathrm{2}} =\mathrm{4} \\ $$$${x}^{\mathrm{2}} \left({x}−\mathrm{1}\right)=\mathrm{4} \\ $$$$\Rightarrow{x}=\mathrm{2}\Rightarrow{a}=\pm\sqrt{\mathrm{2}} \\ $$$$\Rightarrow{y}=\mathrm{2}\Rightarrow{b}=\pm\sqrt{\mathrm{2}} \\ $$
Commented by mathdanisur last updated on 13/Jul/21
cool Ser thanks, y=±(√2)⇒b=±(√2).?
$${cool}\:{Ser}\:{thanks},\:{y}=\pm\sqrt{\mathrm{2}}\Rightarrow{b}=\pm\sqrt{\mathrm{2}}.? \\ $$

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