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# help

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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