Square root and Cube root in the same equation

Question

In a recent test I took, I received a question in an awkward form: $$\sqrt y-\sqrt[3]{1000-y}=16$$ How would I go about solving this?

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score 3 · Answer 1 · answered Mar 23 '19 at 22:54

3

Let $x = \sqrt{y}$ and then you have that $$(x-16)^3 = 1000-x^2$$ which gives you a cubic that you can solve using standard methods.

answered Mar 23 '19 at 22:54

Student

I have tried this, and it leads me to $x^3 - 47 x^2 + 768 x - 5096 = 0$, which according to WolframAlpha doesn't have an integer solution. – Kavi Agrawal Mar 23 '19 at 23:00
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@KaviAgrawal: That is correct, it doesn't have an integral solution. Why do you think it should? – Ross Millikan Mar 23 '19 at 23:04
How would I be able to derive the solution by hand, in that case? – Kavi Agrawal Mar 23 '19 at 23:06
Newton Ralphson is one iterative method that could be used to approximate roots. – Student Mar 23 '19 at 23:07

1 Answers1