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Question Number 39851 by Raj Singh last updated on 12/Jul/18

Commented by MrW3 last updated on 12/Jul/18

$$letu=\sqrt{x}⩾0,v=\sqrt{y}⩾0$$$$u+v=13$$$$u^2+v^2=13$$$$u^2+v^2+2uv-2uv=13$$$${(u+v)}^2-2uv=13$$$${13}^2-2uv=13$$$$\Rightarrow uv=\frac{{13}^2-13}{2}=78$$$$u^2+v^2-2uv+2uv=13$$$${(u-v)}^2+2uv=13$$$${(u-v)}^2=13-2\times 78=-143<0\Rightarrow norealsolution!$$$$$$$$u-v=\pm \sqrt{143}i$$$$\Rightarrow u=\frac{(u+v)+(u-v)}{2}=\frac{13\pm \sqrt{143}i}{2}$$$$\Rightarrow v=\frac{(u+v)-(u-v)}{2}=\frac{13\mp \sqrt{143}i}{2}$$$$\Rightarrow x=u^2=\frac{{13}^2-143\pm 2\times 13\sqrt{143}i}{4}=\frac{13}{2}(1\pm \sqrt{143}i)$$$$\Rightarrow x=v^2=\frac{{13}^2-143\mp 2\times 13\sqrt{143}i}{4}=\frac{13}{2}(1\mp \sqrt{143}i)$$

Answered by tanmay.chaudhury50@gmail.com last updated on 12/Jul/18

$$\sqrt{x}+\sqrt{13-x}=13$$$$x+13-x+2\sqrt{13x-x^2}=169$$$$\sqrt{13x-x^2}=78$$$$13x-x^2=6084$$$$x^2-13x+6084=0$$$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$$$x=\frac{13\pm \sqrt{169-4\times 6084}}{2}$$$$$$$$=\frac{13\pm \sqrt{-24267}}{2}=\frac{13\pm i\sqrt{24167}}{2}$$$$x=\frac{13+i\sqrt{24167}}{2}y=\frac{13-i\sqrt{24167}}{2}$$$$x=\frac{13-i\sqrt{24167}}{2}y=\frac{13+i\sqrt{24267}}{2}$$

Answered by MJS last updated on 12/Jul/18

$$\left(1\right)y={(\sqrt{x}-13)}^2=x+26\sqrt{x}+169$$$$\left(1\right)\mathrm{in}\left(2\right):$$$$x-13\sqrt{x}+78=0$$$$\sqrt{x}=\frac{13}{2}\pm \frac{\sqrt{143}}{2}\mathrm{i}\Rightarrow x=\frac{13}{2}\pm \frac{13\sqrt{143}}{2}\mathrm{i}\Rightarrow$$$$\Rightarrow y=\mathrm{conj}\left(x\right)=\frac{13}{2}\mp \frac{13\sqrt{143}}{2}\mathrm{i}$$

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