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#1 2025-02-27 12:56:04
- rossrossolimo
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Complex Quadratic
I cannot find the answer to this question! Could someone point me in the right direction?
13ix^2 + x - 101=0
#3 2025-02-27 21:33:31
- Bob
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Re: Complex Quadratic
hi rossrossolimo
Welcome to the forum.
The quadratic formula will still work here.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob 
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#4 2025-02-27 23:03:04
Re: Complex Quadratic
The quadratic formula will still work here.
How will it work here? I tried it and I couldnt simplify it.
"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer
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#5 2025-03-01 22:24:44
- Bob
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Re: Complex Quadratic
I've now tried it and it is very 'messy'.
a = 13i b = 1 c = -101
Is that sufficient for an answer? I checked with Wolfram alpha and that's the answer it gave.
If you've got another hour or so to spare, it is possible to find the complex square root of any number using De Moivre's theorem.
First find the modulus and argument of the complex number.
A square root will have a modulus that is the square root of the above mod. and an argument that is half the argument of the above.
That's where it gets messy. I had hoped it would come out to a 'nice, easy' number but it doesn't.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#6 2025-03-02 22:59:15
Re: Complex Quadratic
I first removed the complex number and then tried to find x. Can you please explain the De Moivre's theorem?
"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer
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#7 2025-03-02 23:56:49
- KerimF
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Re: Complex Quadratic
A + iB could be seen as the sum of two vectors on two perpendicular axes.
A on the real axis (horizontal).
B on the imaginary axis (vertical).
The modulus = the length of the sum vector = sqrt(A^2 + B^2) = r
The argument = its angle with the horizontal axis = arctan(B/A) = θ
The new form is usually written as:
r*e^iθ
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#8 2025-03-03 01:21:22
- Bob
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Re: Complex Quadratic
Can you please explain the De Moivre's theorem?
Yes, but now I come to do so I've realised that I was using something related but not de Moivre's theorem itself.
I'll do my formula* first. * I'm not claiming it's named for me, but just that it's what I was using.
Suppose you have a complex number a + ib.
r is called the modulus of the complex number and theta its argument. You can move between the a + ib version of the complex number and the modulus argument version using
Now suppose you have a second complex number with modulus s and argument phi and multiply them together
Thus, for any two complex numbers the modulus of the product is the product of the moduli and the argument of the product is the sum of the arguments.
Bob
Would you like to see the real de moivre also?
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#10 2025-03-03 02:54:38
- Bob
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Re: Complex Quadratic
Ok; using what I have just proved:
and if I multiply that result by another of the same:
and so on. de Moivre's theorem is the general result:
It might not seem like much but it is really useful when dealing with complex number on the Argand Diagram.
eg. Let's say I want to find all three cube roots of 1.
Mod(1) is 1 so all three must have mod 1 too. So, on an Argand Diagram, draw a circle radius 1 around (0,0).
We know that a cube root is 1 but complex theory says there must be two more. They must lie on the circle or their mod will be wrong. And their arguments must add up to zero (or a multiple of 360. So 120 and 240 will work. That gives
I'll multiply out the first to prove it and leave the other as an exercise for you.
The i terms here cancel leaving
Wow! Getting all the LaTex right here was very tricky.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#13 2025-03-04 00:10:54
- Jai Ganesh
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Re: Complex Quadratic
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#15 2025-03-04 01:02:18
- Bob
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Re: Complex Quadratic
You can mark all real numbers on a horizontal line extending left and right to - ∞ and + ∞
Jean-Robert Argand extended this idea to include the imaginary numbers on an axis at right angles to the real axis.
I heard recently that Decartes ridiculed the idea of such numbers, calling them imaginary as an insult. That's not what I believe however. The imaginary axis is a perfect reflection of the real axis in the line y=x, so I prefer the name to be because it's an image of the real axis.
A complex number can be represented by a point somewhere in the plane of the diagram, having its real part as its x coordinate and its imaginary part as its y coordinate.
I've marked a typical point a + ib with coordinates [rcos(theta), rsin(theta)]
It's modulus (distance from the origin) is r where r^2 = x^2 + y^2 and its argument (angle the line makes with the + x axis) theta.
So far all that is true whatever the scales marked on the axes.
But now look at the scale I've chosen. The picture now shows a circle radius 1, centred on the origin. The cube roots of 1 are three points at angle theta = 0°, 120° and 240°.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#17 2025-03-04 05:39:06
- Bob
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Re: Complex Quadratic
Well yes, sort of correct. But 'only' here does not do justice to all the applications that stem from that.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#19 2025-03-05 01:40:57
- Bob
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Re: Complex Quadratic
There's enough here https://en.wikipedia.org/wiki/Complex_number to keep you busy for a few weeks. The applications are about 3/4 of the way down the article with links if you want more details.
What I particularly like is the completeness of the complex numbers. Any algebraic equation has a solution in complex numbers. It also leads to my favourite equation.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#20 2025-03-05 01:49:57
- Bob
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Re: Complex Quadratic
There's enough here https://en.wikipedia.org/wiki/Complex_number to keep you busy for a few weeks. The applications are about 3/4 of the way down the article with links if you want more details.
What I particularly like is the completeness of the complex numbers. Any algebraic equation has a solution in complex numbers. It also leads to my favourite equation.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#22 2025-03-05 23:47:59
- Bob
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Re: Complex Quadratic
I was, of course, hoping you'd ask that.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#23 2025-03-06 03:29:40
Re: Complex Quadratic
I also like that one. 
Euler's equation. Also the god equation but I dont know how to use it. 
Last edited by ktesla39 (2025-03-06 03:33:16)
"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer
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