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Question Number 12023 by tawa last updated on 09/Apr/17

$$\mathrm{Solve}\mathrm{simultaneously}$$$$\mathrm{x}+\mathrm{y}-\sqrt{\mathrm{xy}}=3..........\left(\mathrm{i}\right)$$$$\sqrt{\mathrm{x}+1}+\sqrt{\mathrm{y}+1}=4..........\left(\mathrm{ii}\right)$$

Answered by Mr Chheang Chantria last updated on 10/Apr/17

$$\left(\mathrm{i}\right).\mathit{x}+\mathit{y}-\sqrt{\mathit{x}\mathit{y}}⩾\mathit{x}+\mathit{y}-\frac{\mathit{x}+\mathit{y}}{2}=\frac{\mathit{x}+\mathit{y}}{2}$$$$3⩾\frac{\mathit{x}+\mathit{y}}{2}$$$$\iff \mathit{x}+\mathit{y}⩽6$$$$.\sqrt{\mathit{x}+1}+\sqrt{\mathit{y}+1}⩽\sqrt{2(\mathit{x}+\mathit{y}+2)}$$$$⩽\sqrt{2(6+2)}=4$$$$\Rightarrow \sqrt{\mathit{x}+1}+\sqrt{\mathit{y}+1}⩽4$$$$\mathit{b}\mathit{y}\left(\mathit{i}\mathit{i}\right)\mathit{t}\mathit{h}\mathit{e}\mathit{s}\mathit{i}\mathit{g}\mathit{n}\mathit{e}\mathit{q}\mathit{u}\mathit{a}\mathit{l}\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathit{x}=\mathit{y}=3$$

Commented by Mr Chheang Chantria last updated on 10/Apr/17

$$\mathit{t}\mathit{h}\mathit{i}\mathit{s}\mathit{i}\mathit{s}\mathit{f}\mathit{o}\mathit{r}\mathit{s}\mathit{t}\mathit{u}\mathit{d}\mathit{e}\mathit{n}\mathit{t}.$$$$\mathit{w}\mathit{e}\mathit{h}\mathit{a}\mathit{v}\mathit{e}{(\sqrt{\mathit{x}}-\sqrt{\mathit{y}})}^2⩾0\mathrm{for}\mathit{x},\mathit{y}⩾0$$$$\mathit{x}-2\sqrt{\mathit{x}\mathit{y}}+\mathit{y}⩾0$$$$\mathit{x}+\mathit{y}⩾2\sqrt{\mathit{x}\mathit{y}}$$$$\mathit{o}\mathit{r}\frac{\mathit{x}+\mathit{y}}{2}⩾\sqrt{\mathit{x}\mathit{y}}$$$$\mathit{o}\mathit{r}-\frac{\mathit{x}+\mathit{y}}{2}⩽-\sqrt{\mathit{x}\mathit{y}}$$$$$$

Commented by tawa last updated on 10/Apr/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Answered by ajfour last updated on 09/Apr/17

$$squaring...\left(ii\right)$$$$x+y+2+2\sqrt{(x+2)(y+1)}=16$$$$3+\sqrt{xy}+2+2\sqrt{(x+1)(y+1)}=16$$$$2\sqrt{(x+1)(y+1)}=11-\sqrt{xy}$$$$squaring:$$$$4(xy+1+x+y)=121+xy-22\sqrt{xy}$$$$4(xy+1+3+\sqrt{xy}=121+xy-22\sqrt{xy}$$$$3xy+26\sqrt{xy}-105=0$$$$if\sqrt{xy}=t$$$$3t^2+26t-105=0$$$$t=\sqrt{xy}=\frac{-26+\sqrt{676+1260}}{6}$$$$\sqrt{xy}=\frac{-26+44}{6}=3....\left(iii\right)$$$$xy=9$$$$from....\left(i\right)$$$$x+y=3+\sqrt{xy}$$$$\Rightarrow x+y=6....\left(iv\right)$$$${(x-y)}^2={(x+y)}^2-4xy$$$$so,{(x-y)}^2=36-4\left(9\right)=0$$$$\Rightarrow x=y$$$$andasx+y=6$$$$x=y=3.$$

Commented by tawa last updated on 09/Apr/17

$$\mathrm{wow}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 09/Apr/17

$$x+1+y+1+2\sqrt{x+1}\sqrt{y+1}=16$$$$\sqrt{xy}+5+2\sqrt{x+1}\sqrt{y+1}=16$$$$2\sqrt{x+1}\sqrt{y+1}=11-\sqrt{xy}$$$$4(xy+x+y+1)=121-22\sqrt{xy}+xy$$$$x+y=t$$$$4[{(t-3)}^2+t+1]=121-22(t-3)+{(t-3)}^2$$$$3{(t-3)}^2+26(t-3)-105=0$$$$t-3=\frac{-26\pm \sqrt{{26}^2+4\times 3\times 105}}{2\times 3}=\frac{-26\pm 44}{6}=3,-11.66$$$$t=x+y=6\Rightarrow \sqrt{xy}=3\Rightarrow xy=9$$$$\Rightarrow x=y=3◼$$$$$$$$$$

Commented by tawa last updated on 09/Apr/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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