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Find the real roots of an equation.

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,756
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

Well-known member
MHB Math Helper
Jan 26, 2012
1,404
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

Well-known member
MHB Math Helper
Jan 29, 2012
661
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?
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,756

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
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

Well-known member
Feb 3, 2013
134
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 ..
 
Last edited:

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661