MHB Solving \[x^{4}-4x^{3}+10=0\] with a "Binary Search

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To solve the equation \(x^{4}-4x^{3}+10=0\), users suggest employing numerical methods such as binary search or Newton's Method for approximating roots. While binary search provided some success, alternatives like using Wolfram Alpha for exact solutions are recommended. The rational root theorem is mentioned but noted as ineffective for this specific equation. The discussion highlights the challenges of solving quartic equations and the need for numerical approximation techniques. Overall, utilizing computational tools or established numerical methods is advised for finding solutions.
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Hello all

I am trying to draw a graph of a function. On the way, I wanted to see where the function meet the x axis, so I put y=0. It gave me this:

\[x^{4}-4x^{3}+10=0\]

How do I solve this equation ?

Thanks !

I tried a "binary search" and got really close to the answer, but I guess there must be a better way...
 
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Yankel said:
Hello all

I am trying to draw a graph of a function. On the way, I wanted to see where the function meet the x axis, so I put y=0. It gave me this:

\[x^{4}-4x^{3}+10=0\]

How do I solve this equation ?

Thanks !

I tried a "binary search" and got really close to the answer, but I guess there must be a better way...

It's not easy...
Quartic function - Wikipedia, the free encyclopedia

A binary search or goal seek in excel are probably your best bets with a numerical approximation being next. Honestly this is one of those questions I'd just put into wolfram and use their answer(s)
 
Yankel said:
Hello all

I am trying to draw a graph of a function. On the way, I wanted to see where the function meet the x axis, so I put y=0. It gave me this:

\[x^{4}-4x^{3}+10=0\]

How do I solve this equation ?

Thanks !

I tried a "binary search" and got really close to the answer, but I guess there must be a better way...
You can occasionally use the rational root theorem, but it doesn't work for this equation.

-Dan
 
I understand, thank you !

What numerical approximation methods do we have to solve such equations ?
 
Yankel said:
I understand, thank you !

What numerical approximation methods do we have to solve such equations ?

Perhaps the best known is:

Newton's Method
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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