This YouTube https://www.youtube.com/watch?v=_Y778c5g3zU video goes through the steps of finding all the solutions for a sixth-order equation, name (x-2)6 = 729. As expected there are two real solutions and four complex solutions. However, when I went to Desmos | Graphing Calculator and entered that, it gives two vertical lines at the two real solutions and thus an infinite number of complex solutions. Is Desmos faulty or at least limited in the degree of equations it can plot, or am I doing something wrong?
It’s not clear to me what that calculator is plotting.
The vertical lines are at x= -1 and x=5, the two obvious real roots, so it’s displaying something relevant to the equation. But not the discrete solutions the video showed.
That doesn’t look like a complex plane graph, so I imagine the grapher discarded the complex roots.
As above, graphing calculators show the real plane, not the complex plane. And in the real plane, the only valid solutions are x=-1 and x=5, which graph as vertical lines.
They aren’t actually vertical lines. If you adjust the scale (using the wrench button in the top right corner) to -750<y<50 you will see the function is “U” shaped.
~Max
Incidentally, just caught the YouTube video. It’s nice, but way too complicated a method for my tastes.
I prefer a more geometric approach as per Euler.
If we assume a function mapping complex numbers, then there are 6 solutions, as noted. They lie at the vertices a regular hexagon of outer radius 3 on the complex plane centered at x=2.
So, there are the solutions (-1,0) and (5,0) plus 4 others are the other points of the hexagon which would be +/- 60 degrees off the real axis using (2,0) as the vertex.
And so we basically just need to know/remember a bit of trig and that cos(60 deg) = cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2 to find the rest (to be fair, not everybody may know or remember these).
With that knowledge, two more solutions would be given by 2+3cos(pi/3) +/- i3sin(pi/3), which is [7 +/- i3*sqrt(3)]/2.
And the two remaining solutions at 2+3cos(2pi/3) +/- i3sin(2pi/3), which is [1 +/- i3*sqrt(3)]/2
If they’re not, then there really is an issue with the software. They should be vertical lines.
My mistake, I had typed it as a function of x. f(x)=(x-2)^6 - 729 is “U” shaped.
~Max
Yeah, that makes sense then. Solutions given where it crosses the x-axis, i.e. where f(x) = 0
I’m not sure I understand the question. The solutions are obviously x-2 = 3*u where is u is any of the six sixth roots of 1, that is
\pm1, \frac{\pm1\pm\sqrt{3}}2
The OP did not realize graphic calculators do not plot complex valued solutions and thought there may have been a bug.
Even if you type it in as the equation (x-2)^6 = 729, Desmos interprets that as your wanting to graph y = (x-2)^6 - 729. Not unreasonable, as the solutions of the equation are where the graph of the function crosses the x-axis.
Here’s a plot of the solutions:
Exactly as @Great_Antibob described.
And here in Desmos,
~Max
No, that’s not what Desmos is doing. When you type an equation with a single variable, Desmos assumes that you’re defining that variable. So it interprets “(x-2)^6 = 729” as “x = -1 and x = -5”. And “x=-1” is a straight vertical line, and so is “x=5”.
EDIT: Here’s a demonstration with a simpler equation
You’re right, I’m wrong. I thought it was giving me the same graph as the function, but now I can’t reproduce that.
Even simpler would be an equation like “x = 2.” The graph of an equation like that in the xy-plane is a vertical line, because it’s made up of all the points (x, y) which satisfy that equation.
So it only makes sense that if you type in the equation (x-2)^6 = 729 you would get \{(x, y) \in \mathbb{R}^2 | (x-2)^6 = 729\}, which would consist of those two vertical lines.
Yes, but I was demonstrating that Desmos does implicit definitions, too. I wasn’t surprised the first time I saw it draw a vertical line for “x = 2”, but I was surprised the first time I saw it from a quadratic.
The ability to deal with implicit relations in general is one of Desmos’ greatest strengths, compared to typical graphing calculators. You can put in any equation you want, with any complicated mishmash of x and y that couldn’t possibly be solved for either variable, and it’ll still show you a graph of the set of all points that satisfy the equation.
So is there a way in Desmos of restating the equation that gives the six points of
@Zoobi 's plot?