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Question Number 98192 by behi83417@gmail.com last updated on 12/Jun/20

Commented by bemath last updated on 12/Jun/20

$${(\mathrm{x}+\mathrm{y})}^2-\mathrm{xy}=133$$$$\mathrm{set}\sqrt{\mathrm{xy}}=\mathrm{u}\mathrm{\&}\mathrm{x}+\mathrm{y}=\mathrm{v}$$$$\iff {\mathrm{v}}^2-{\mathrm{u}}^2=133\mathrm{\&}\mathrm{v}+\mathrm{u}=19$$$$(\mathrm{v}-\mathrm{u})(\mathrm{v}+\mathrm{u})=133$$$$19(\mathrm{v}-\mathrm{u})=133\Rightarrow \mathrm{v}-\mathrm{u}=\frac{133}{19}=7$$$$\mathrm{we}\mathrm{get}\{\begin{array}{l}\mathrm{v}=13\\ \mathrm{u}=6\end{array}$$$$\{\begin{array}{l}\mathrm{xy}=36\\ \mathrm{x}+\mathrm{y}=13\end{array}$$$$(\mathrm{x},\mathrm{y})=\{\begin{array}{l}(4,9)\\ (9,4)\end{array}$$$$$$

Answered by mr W last updated on 12/Jun/20

$$v=\sqrt{xy}$$$$u=x+y$$$$x^2+y^2+2xy-xy=133$$$$u^2-v^2=133$$$$\Rightarrow (u+v)(u-v)=133$$$$u+v=19$$$$\Rightarrow u-v=\frac{133}{19}=7$$$$\Rightarrow u=x+y=\frac{19+7}{2}=13$$$$\Rightarrow v=\frac{19-7}{2}=6$$$$\Rightarrow xy=v^2=36$$$$x,yarerootsof$$$$t^2-13t+36=0$$$$t=\frac{13\pm 5}{2}=4,9$$$$\Rightarrow (x,y)=(4,9)$$

Answered by MJS last updated on 12/Jun/20

$$\mathrm{I}\mathrm{can}\mathrm{see}x=4\wedge y=9\vee x=9\wedge y=4\mathrm{but}\mathrm{how}$$$$\mathrm{to}\mathrm{solve}\mathrm{it}?$$$$x=u-v\wedge y=u+v$$$$\{\begin{array}{l}3u^2+v^2-133=0\\ 2u-19+\sqrt{u^2-v^2}=0\end{array}$$$$\{\begin{array}{l}{v}^2=-3u^2+133\\ v^2=-3u^2+76u-361\end{array}\Rightarrow u=\frac{13}{2}\wedge v=\pm \frac{5}{2}$$

Commented by behi83417@gmail.com last updated on 12/Jun/20

$$\mathrm{thanks}\mathrm{to}\mathrm{all}\mathrm{my}\mathrm{masters}.$$

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