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Find the largest possible value of \(x \) in the simplified form  \(x=\frac{a+b\sqrt{c}}{d}\)  if  \(\frac{5x}{6}+1=\frac{3}{x}\) , where a,b,c and d are integers. What is   \(\frac{acd}{b}\)?

 Aug 29, 2022
edited by Alan  Aug 30, 2022
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First find the value of \(x.\) Expanding and combining like terms, we find that it becomes a quadratic equation, \(5x^2+6x-18=0\).

 

Solving using the quadratic formula, we see that \(x\) has two solutions- \(\frac{-3\pm3\sqrt{11}}{5}\). We want the largest possible solution, so the \(\pm\) becomes a \(+\) instead.

 

Thus we have \(\frac{-3+3\sqrt{11}}{5}=\frac{a+b\sqrt{c}}{d}\). Substituting these values in for each other, we find that \(a=-3, b=3, c=11, d=5.\) So the value of \(\frac{acd}b=\frac{-3\cdot11\cdot5}{3}=\boxed{-55}.\)

 Sep 4, 2022

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