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Given$$x+y=2010,\\(\sqrt[3]{x}+\sqrt[3]{y})(\sqrt[3]{x}+11)(\sqrt[3]{y}+11)=2942,$$ find the value of $\sqrt[3]{x}+\sqrt[3]{y}$.

My attempts:

Let $u=\sqrt[3]{x} , v=\sqrt[3]{y}$.

$(u+v)(u^2-uv+v^2)=2010 \tag{1}$

$(u+v)(uv+11u+11v+121)=2942 \tag{2}$

$(1)+(2)$:

$$(u+v)(u^2+v^2+11u+11v+121)=4952$$

How to proceed?

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  • 3
    $\begingroup$ Do yourself a favor and switch variables to $t=\sqrt[3]x$ and $u=\sqrt[3]y$ before you dive too deep into the algebra. $\endgroup$ – Henning Makholm Apr 19 '17 at 11:25
  • $\begingroup$ Are $x$ and $y$ known to be integers? $\endgroup$ – Henning Makholm Apr 19 '17 at 11:41
  • $\begingroup$ Problem statement don't mention on this matter. I think $x, y \in \mathbb{R}$. $\endgroup$ – carat Apr 19 '17 at 11:44
  • $\begingroup$ ${(u+v)}=\frac {2010} { (u^2-uv+v^2)}$ and ${(u+v)}=\frac {4592} {(u^2+v^2+11u+11v+121)}$ so $\frac {2010} { (u^2-uv+v^2)}=\frac {4592} {(u^2+v^2+11u+11v+121)}$ looks like an obvious strategy, did you try that already? Seems unlikely to help, but worth looking at. $\endgroup$ – Arby Apr 19 '17 at 11:46
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From what you've done, we have that $$(1)\iff 2010=(u+v)^3-3uv(u+v)\tag3$$ and that $$(2)\iff uv(u+v)=2942-11(u+v)^2-121(u+v)\tag4$$

From $(3)(4)$, we have $$2010=(u+v)^3-3(2942-11(u+v)^2-121(u+v)),$$ i.e. $$t^3+33t^2+363t-10836=0\tag5$$ where $t=u+v$.

Now checking if a divisor of $10836=2^2\times 3^2\times 7\times 43$ is a solution, we see that $t=12$ is a solution to have $$(5)\iff (t-12)(t^2+45t+903)=0$$ where $t^2+45t+903\gt 0$ since $45^2-4\cdot 1\cdot 903\lt 60^2-4\times 900=0$.

It follows from this that $$\sqrt[3]{x}+\sqrt[3]{y}=\color{red}{12}$$

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  • $\begingroup$ Yep, same approach as my solution, just factorised a bit more clearly. Nice. $\endgroup$ – stanley dodds Apr 19 '17 at 11:59
  • $\begingroup$ Thank you very much, mathlove, stanley dodds. $\endgroup$ – carat Apr 19 '17 at 12:22
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u and v look quite similar, so I will just use a and b instead: $$3(a+b)(ab+11a+11b+121)+(a+b)(a^2-ab+b^2)=3\times2942+2010=10836$$ $$(a+b)(a^2+2ab+b^2+33a+33b+363)=10836$$ $$(a+b)((a+b)^2+33(a+b)+363)=10836$$ $$(a+b)^3+33(a+b)^2+363(a+b)-10836=0$$ Solving for $a+b$, the real root to this is $a+b=12$, so the answer is 12. I'm not sure if this is the desired approach, but it works.

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