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# How do you find out the soloutions of cubic equations without a calculator?

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How do you solve the cubic equations like 8x^3+4x^2-4x-1=0 and y^3+4y^2+3y-1=0 withour a calculator? Thank you!

Guest Apr 18, 2015

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I have added this info to our reference material sticky thread  (In the calculator section)

Melody  Apr 18, 2015
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If it's an exam question, often it has been designed so you'll be able to see one solution by inspection, e.g., try various easy possibilities such as ½, ¼, 1, 2, 3, 5, 10, and their negatives. Once you know one solution, call it k, then use division by (x-k) to end up with a quadratic and solve that using the memorized formula.

If the problem hasn't been designed for easy solution, then I'm afraid you're SOOL.

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\$\$\\8x^3+4x^2-4x-1=0\\\$\$

this one is not jumping out at me.

Here is the graph showing the 3 real solutions

https://www.desmos.com/calculator/xhxudxv72l

and

\$\$y^3+4y^2+3y-1=0\$\$

this one is not jumping out at me either.

https://www.desmos.com/calculator/b5cmcqjhv0

You could use Newton's method of approximating roots to get the answers.   :)

Melody  Apr 18, 2015
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How do you solve the cubic equations like 8x^3+4x^2-4x-1=0 and y^3+4y^2+3y-1=0 withour a calculator?

With some difficulty!  See  http://en.wikipedia.org/wiki/Cubic_function, for example,  for how to do it.

The solution to your first example (done with a calculator!) is shown below. It's a screen shot of part of the Microsoft Mathematics 4 calculator, which is a free download.

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Alan  Apr 18, 2015
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Thanks Alan,

I have seen you use that calc a few times.  I should check it out :)

Melody  Apr 18, 2015
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I've only recently come across this calculator, so I've been trying it out.  It's ok, though it does have a few annoying features (for example, changing the colour of lines on a graph is cumbersome), and it isn't powerful enough for general "industrial" use, but it should be very useful in an educational environment.  One useful feature, for example, is that after you solve some equations you have the option to examine the solution steps  as well as just see the results (though this option isn't offered for the cubic solution above I notice!).

Alan  Apr 18, 2015
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Okay so it might sometimes be a good alternative to the paid version of Wolfram|Alpha.

There were two download versions.  I wasn't sure what the difference was.  I just chose the 64 one :/

Melody  Apr 18, 2015
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The 64 one will only work on 64-bit computers.  The other one will work on both 32 and 64-bit computers.

Alan  Apr 18, 2015
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Yes that sounds logical Alan but the other one was not 32 bit  it was 90 something :/

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For a long time now I have wanted a graphing program where I could give some info and get a suitable graph generated.  Maybe without any numbers on the x or y axis.

I often just want to display the shape of the graph.

Like

graph ax^3+bx^2+cx+d     a>0         [no axes]

or

graph ax^3+bx^2+cx+d     a>0  through  (0,3)(0,-2)and (20,7)

If anyone sees software like this I would REALLY like to know about it!

Melody  Apr 18, 2015
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Well, the Microsoft one can do your first condition (see below), though I'm not sure about your second one (do you mean you want the calculator to find a curve that passes through the specified points and then plot the curve?).

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Alan  Apr 18, 2015
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"do you mean you want the calculator to find a curve that passes through the specified points and then plot the curve?"

There are many alternate things that I want but yes,

If I could give the basic shape with a general formula and then add whatever other criteria that I wanted, that would be great.

Desmos would be able to do some of it if I had the option of NOT displaying the axes and/or  NOT display any numbers on the axes.

I assume with your graph you just did not display the axes?

Melody  Apr 18, 2015
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You can use the Solve equations function to find the coefficients of a polynomial that fits through a number of points and then plot the resulting function in MM4.

For example, suppose you wanted to fit y = a.x2 + b.x + c through the points (-2, 3), (1, 6) and (2, 15)

You can do this with MM4 (though, annoyingly, you have to use x1, x2 and x3 for the unknown coefficients, whereas you really would like to use a, b and c so as not to confuse the unknowns with values on the x-axis).

Having done this you can plot the graph of the resulting function (which you have to construct manually, now you know the coefficients):

I didn't bother turning off the axes in the above, although that can be done of course.

Alan  Apr 18, 2015
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That is really confusing Alan but I think I finally understand what you have done.

It would be very slow to use this on a regular basis but maybe practice would increase speed:/

It would be SOOO much easier to understand and do if the constant coefficients were just a,b,c etc

It amazes me sometimes that people can be so very clever to program these things and yet they make the most obvious errors or complications.   :/

Thanks very much for taking the time to look at this for me :))

Melody  Apr 18, 2015
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