Find the real roots of an equation.

[anemone]

Hello to all members of the forum,

Problem:
Find the real roots of the equation
$\displaystyle x^3+2ax+\frac{1}{16}=-a+\sqrt{a^2+x-\frac{1}{16}}\;\;\; (0<a<\frac{1}{4}) $

I really have no idea on how to even start to work with this problem, could anyone please help me?

Thanks.

 

[Prove It]

Hello to all members of the forum,

Problem:
Find the real roots of the equation
$\displaystyle x^3+2ax+\frac{1}{16}=-a+\sqrt{a^2+x-\frac{1}{16}}\;\;\; (0<a<\frac{1}{4}) $

I really have no idea on how to even start to work with this problem, could anyone please help me?

Thanks.

You should probably start with

$\displaystyle \begin{align*} x^3 + 2ax + \frac{1}{16} &= -a + \sqrt{a^2 + x - \frac{1}{16} } \\ x^3 + 2ax + \frac{1}{16} + a &= \sqrt{a^2 + x - \frac{1}{16}} \\ \left( x^3 + 2ax + \frac{1}{16} + a \right)^2 &= a^2 + x - \frac{1}{16} \end{align*}$

Now expand everything, and try to solve the resulting 6th order polynomial equation if possible...

 

[Fernando Revilla]

Problem:
Find the real roots of the equation
$\displaystyle x^3+2ax+\frac{1}{16}=-a+\sqrt{a^2+x-\frac{1}{16}}\;\;\; (0<a<\frac{1}{4}) $

Hello anemone,

Could you provide the source and/or context? Is it necessary a closed form solution?

 

[anemone]

Hello anemone,

Could you provide the source and/or context? Is it necessary a closed form solution?

Hi Fernando,

The problem comes from here:
The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary ... - Давид Оскарович Шклярский, &#

(Question 169)

Thanks.

 

[Fernando Revilla]

The problem comes from here:
The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary ... - Давид Оскарович Шклярский, &#
(Question 169)

A little story: yesterday I spend two hours tryng to solve the problem without success, so I had a suspection. Browsing on the net I saw the solution in a pdf. There is a typo, in the left side the first term is $x^2$ instead of $x^3$. Have a look at the following link

http://www.fernandorevilla.es/wp-content/uploads/2013/02/olympiad.pdf

There are at least two ways of solving the problem. The first way (I tried in the same way with $x^3$) using a graphical method, the second one using an horrible method. Ask your doubts.

 

[zzephod]

Hello to all members of the forum,

Problem:
Find the real roots of the equation
$\displaystyle x^2+2ax+\frac{1}{16}=-a+\sqrt{a^2+x-\frac{1}{16}}\;\;\; (0<a<\frac{1}{4}) $

I really have no idea on how to even start to work with this problem, could anyone please help me?

Thanks.

Using Fernando Revilla's correction to the problem, and hoping my algebra is working properly today:

... forget it, it does not work ..

 

[Fernando Revilla]

... forget it it does not work ..

At any rate, thanks for your contribution and welcome to MHB.