Solving x (NOT SOLVED YET)

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BenjaminGC

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How do I solve x without using a computer, just oldschool/analog? I mean for the whole of f(x) and not only the encircled part.12644
 
This ends up as a 6th degree polynomial to find roots for.

There are no closed form solutions to that.
 
What method/s you have been taught to calculate the roots of a function?
 
As several people have indicated, it is impossible to help you without knowing exactly what the problem is and without knowing what you know about finding roots. It is not going to help you if we suggest Newton's method for finding a solution or approximate solution if you do not know calculus.

I shall assume you know no calculus. I shall also assume that you have already tried the rational root theorem and found it inapplicable. And finally I shall assume that is 1 rather than 4 in the denominator.

[MATH]f(x) = 2x + \dfrac{\sqrt{x + 3}}{x^2 - 1} = \dfrac{2x^3 - 2x + \sqrt{x + 3}}{x^2 - 1}.[/MATH]
You can find out a lot about the general location of the zeroes using simple algebraic concepts.

What is the domain of this function?

Does this function have asymptotes? If so where and what kind?

Where does the function increase?

Where does the function decrease?

Using the answers to these and similar questions, can you determine how many real zeroes there are and where they approximately are to be found?

Example. If x > 1, the function is necessarily positive so there is no zero if x > 1.
 
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Otis said:
I had thought it was 7.

?
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Oh that is a possibility as well.

I wish I knew what the problem is. It looks hard for a student of algebra so I thought some hints were in order.

o_O
 
BenjaminGC said:
...she said, equal it to 0 ...
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I sincerely hope this is not an accurate quotation.

Graph [math]y\;=\;-2x\;and\;y\;=\;\dfrac{\sqrt{x-3}}{x^{2}-1}[/math] individually on the same set of coordinate axes.
 
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tkhunny said:
… Graph [math]y\;=\;-2x\;and\;y\;=\;\dfrac{\sqrt{x-3}}{x^{2}-1}[/math] …
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That radicand should be x+3, and the OP asked about old-school methods (i.e., no technology).

?
 
JeffM said:
OMG. Do you suppose that all that was meant was

[MATH]\text {What is the numeric value of } f(0)?[/MATH]
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LOL. I'm thinking the class may be using technology to find some x-intercepts and the OP asked about paper-n-pencil methods later out of curiosity.

Anyway, the teacher asked for "the value of x", so I'm pretty sure they're talking about the single root. Your suggestion in post #8 (narrow down the interval) is a good start. Numerical grunt work to further narrow the interval about the root works, after that. More effort leads to better approximations.

I hope the OP is willing to use a scientific calculator evaluating those radicals, at least.

;)
 
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Otis said:
LOL. I'm thinking the class may be using technology to find some x-intercepts and the OP asked about paper-n-pencil methods later out of curiosity.

Anyway, the teacher asked for "the value of x", so I'm pretty sure they're talking about the single root. Your suggestion in post #8 (narrow down the interval) is a good start. Numerical grunt work to further narrow the interval about the root works, after that. More effort leads to better approximations.

I hope the OP is willing to use a scientific calculator evaluating those radicals, at least.

;)
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Of course, before computers and calculators we had tables of logarithms and slide rules, which students today are not taught how to use. So modern students do not realize that our paper-and-pencil methods were more powerful than theirs. It is hard for the young to realize that progress involves both gain and loss.

This is not a plea to teach the older methods; technology has made them obsolete for any practical purpose. But it might be worth pointing out to modern students that their forebearers had tools that they do not.
 
Otis said:
That radicand should be x+3, and the OP asked about old-school methods (i.e., no technology).

?
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Fair enough. Typos happen.

When I first solved such a problem, I used a pencil and some paper to draw the graphs. I was talking OLD SCHOOL. How old qualifies?
 
tkhunny said:
Fair enough. Typos happen.

When I first solved such a problem, I used a pencil and some paper to draw the graphs. I was talking OLD SCHOOL. How old qualifies?
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The problem is that sketching curves is greatly helped by calculus, and I doubt this student knows any.
 
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