Question

tìm 3 số a, b, c dương biết : \(\dfrac{\sqrt{ab}-1}{3}=\dfrac{\sqrt{bc}-3}{9}=\dfrac{\sqrt{ca}-5}{-6}\) và \(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}=11\)

Answer 1

Lời giải:

Đặt \(\frac{\sqrt{ab}-1}{3}=\frac{\sqrt{bc}-3}{9}=\frac{\sqrt{ca}-5}{-6}=t\)

\(\Rightarrow \left\{\begin{matrix} \sqrt{ab}=3t+1\\ \sqrt{bc}=9t+3\\ \sqrt{ca}=5-6t\end{matrix}\right.\)

\(\Rightarrow \sqrt{ab}+\sqrt{bc}+\sqrt{ca}=6t+9\)

\(\Leftrightarrow 11=6t+9\Leftrightarrow t=\frac{1}{3}\)

Khi đó : \(\left\{\begin{matrix} \sqrt{ab}=2\\ \sqrt{bc}=6\\ \sqrt{ac}=3\end{matrix}\right.\) \(\Rightarrow \left\{\begin{matrix} ab=4\\ bc=36\\ ac=9\end{matrix}\right.\Rightarrow abc=\sqrt{4.36.9}=36\)

\(\Rightarrow \left\{\begin{matrix} c=\frac{abc}{ab}=9\\ a=\frac{abc}{bc}=1\\ b=\frac{abc}{ac}=4\end{matrix}\right.\)

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