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!
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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this one is not jumping out at me.
Here is the graph showing the 3 real solutions
this one is not jumping out at me either.
You could use Newton's method of approximating roots to get the answers. :)
Do you know about that?
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.
I have seen you use that calc a few times. I should check it out :)
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!).
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 :/
The 64 one will only work on 64-bit computers. The other one will work on both 32 and 64-bit computers.
Yes that sounds logical Alan but the other one was not 32 bit it was 90 something :/
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.
graph ax^3+bx^2+cx+d a>0 [no axes]
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!
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?).
"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?
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.
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 :))