Question-220034

Question Number 220034 by Tawa11 last updated on 04/May/25

Answered by efronzo1 last updated on 04/May/25

(1) x^2 = 96a (2) y^2 = 96b (3) a+b = 96⇒b= 96−a ⇒ 36^2 = ab 36^2 = 96a−a^2 a^2 −96a + 1296 = 0 a = 48+12(√7)= 12(4+(√7) ) b = 48−12(√7) = 12(4−(√7) ) x = (√(96(48+12(√7))))= 24((√7) +1) y = (√(96(48−12(√7) ))) = 24((√7)−1)

$$\:\left(\mathrm{1}\right)\:\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{96}{a} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{96}{b} \\ $$$$\:\left(\mathrm{3}\right)\:{a}+{b}\:=\:\mathrm{96}\Rightarrow{b}=\:\mathrm{96}−{a} \\ $$$$\:\Rightarrow\:\mathrm{36}^{\mathrm{2}} \:=\:{ab}\: \\ $$$$\:\:\:\:\:\:\:\mathrm{36}^{\mathrm{2}} \:=\:\mathrm{96}{a}−{a}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:{a}^{\mathrm{2}} −\mathrm{96}{a}\:+\:\mathrm{1296}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:{a}\:=\:\mathrm{48}+\mathrm{12}\sqrt{\mathrm{7}}=\:\mathrm{12}\left(\mathrm{4}+\sqrt{\mathrm{7}}\:\right) \\ $$$$\:\:\:\:\:\:{b}\:=\:\mathrm{48}−\mathrm{12}\sqrt{\mathrm{7}}\:=\:\mathrm{12}\left(\mathrm{4}−\sqrt{\mathrm{7}}\:\right) \\ $$$$\left.\:\:\:\mathrm{x}\:=\:\sqrt{\mathrm{96}\left(\mathrm{48}+\mathrm{12}\sqrt{\mathrm{7}}\right.}\right)=\:\mathrm{24}\left(\sqrt{\mathrm{7}}\:+\mathrm{1}\right) \\ $$$$\:\:\:\mathrm{y}\:=\:\sqrt{\mathrm{96}\left(\mathrm{48}−\mathrm{12}\sqrt{\mathrm{7}}\:\right)}\:=\:\mathrm{24}\left(\sqrt{\mathrm{7}}−\mathrm{1}\right) \\ $$

Commented by Tawa11 last updated on 04/May/25

$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{appreciate}. \\ $$

Answered by mr W last updated on 04/May/25

xy=36×96 (y/( (√(y^2 −36^2 ))))=(x/(36)) (y^2 /( (√(y^2 −36^2 ))))=((xy)/(36))=96 y^4 −96^2 y^2 +(96×36)^2 =0 ⇒y^2 =((96^2 )/2)±(√((((96^2 )/2))^2 −(96×36)^2 )) ⇒x, y =(√(((96^2 )/2)±(√((((96^2 )/2))^2 −(96×36)^2 )))) =24((√7)±1)

$${xy}=\mathrm{36}×\mathrm{96} \\ $$$$\frac{{y}}{\:\sqrt{{y}^{\mathrm{2}} −\mathrm{36}^{\mathrm{2}} }}=\frac{{x}}{\mathrm{36}} \\ $$$$\frac{{y}^{\mathrm{2}} }{\:\sqrt{{y}^{\mathrm{2}} −\mathrm{36}^{\mathrm{2}} }}=\frac{{xy}}{\mathrm{36}}=\mathrm{96} \\ $$$${y}^{\mathrm{4}} −\mathrm{96}^{\mathrm{2}} {y}^{\mathrm{2}} +\left(\mathrm{96}×\mathrm{36}\right)^{\mathrm{2}} =\mathrm{0}\: \\ $$$$\Rightarrow{y}^{\mathrm{2}} =\frac{\mathrm{96}^{\mathrm{2}} }{\mathrm{2}}\pm\sqrt{\left(\frac{\mathrm{96}^{\mathrm{2}} }{\mathrm{2}}\right)^{\mathrm{2}} −\left(\mathrm{96}×\mathrm{36}\right)^{\mathrm{2}} } \\ $$$$\Rightarrow{x},\:{y}\:=\sqrt{\frac{\mathrm{96}^{\mathrm{2}} }{\mathrm{2}}\pm\sqrt{\left(\frac{\mathrm{96}^{\mathrm{2}} }{\mathrm{2}}\right)^{\mathrm{2}} −\left(\mathrm{96}×\mathrm{36}\right)^{\mathrm{2}} }} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{24}\left(\sqrt{\mathrm{7}}\pm\mathrm{1}\right) \\ $$

Commented by Tawa11 last updated on 04/May/25