Actually this is a little urgent. While I was studying for the exam I encountered with this question. Neither my friends nor I could find the solution. It really seems easy though. Is there a short way or formula that I don't know?
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1What is $a$? (Real, rational, integer?) – flawr Mar 11 '16 at 22:57
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I think $a$ is positive real number. It should suffit. – SiXUlm Mar 11 '16 at 22:58
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It isn't given in the question. @SiXUlm – ece Mar 11 '16 at 23:14
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I guess the solution is 2 because this is what i found. Below you can see the way to the solution. @TakahiroWaki – ece Mar 12 '16 at 00:12
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If $t = \sqrt{a}$, your equation says $t^3 - 7 t - 2 \sqrt{5} = 0$. It's annoying to have that square root in the equation, so let $t = \sqrt{5} u$, and then it becomes $5 u^3 - 7 u - 2 = 0$. This factors as $(u+1)(5 u^2 - 5 u - 2)$, from which you can get the solutions. Caution: you probably don't want negative values of $t$.
Robert Israel
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Thanks! This really helped me. :)) but i'm not sure that they expect this solution from us. If there is a quicker way, i'd really want to learn that, too. – ece Mar 12 '16 at 00:20
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This is as quick as it can get. If there is no rational solution of $A u^3+B u+C=0$ with non-zero integers $A,B,C$ then it gets long. But there is a quick way to find rational roots, if any, when $A.B,C$ are integers.. – DanielWainfleet Mar 12 '16 at 01:00