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Question Number 151391 by mathdanisur last updated on 20/Aug/21

$$\mathrm{if}\mathrm{a};\mathrm{b}\in \mathbb{R}\mathrm{and}$$$$({\mathrm{a}}^2+1)({\mathrm{b}}^2+1)+9=6(\mathrm{a}+\mathrm{b})$$$$\mathrm{find}{\mathrm{a}}^2+{\mathrm{b}}^2=?$$

Answered by dumitrel last updated on 20/Aug/21

$$(a^2+1)(b^2+1)\stackrel{cbs}{⩾}{(a+b)}^2\Rightarrow$$$$6(a+b)⩾{(a+b)}^2+9\Rightarrow {(a+b-3)}^2⩽0$$$$a+b=3\Rightarrow {\left(ab\right)}^2+9-2ab+1=9\Rightarrow ab=1\Rightarrow$$$$a^2+b^2=7$$

Commented by mathdanisur last updated on 20/Aug/21

$$\mathrm{Thank}\mathrm{You}\mathbf{S}\mathrm{er}\mathrm{Cool}$$

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