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Alright, this is going to be another example about factorizing over the complex field.
You can see the work from the previous video.
With this example we will be using the quadratic equation.
So, like the previous
example, we're gonna be using the form ax^2 + bx + c = 0
and the particular example is going to be z^2 + 3z + 5 = 0
um... alright let's just review the quadratic equation really quick. That is,
x = [ -b +/- sqrt( b^2 - 4ac ) ] / 2a
Alright, now this just a matter of plugging in the values.
Now we have x = [ -3 +/- sqrt( 3^2 - (4)(5) ) ] / 2
And we can simplify that to [ -3 +/- sqrt( -11 ) ] / 2
And we can simplify that.
Now,
the definition of 'i'
i^2 equals negative... OK, sorry.
i^2 = -1
So, we take the square root of both sides
and we get the sqrt( i^2 ) = sqrt ( -1 )
By arithmetic, i = sqrt ( -1 )
So, we can just take out sqrt( -1 )
[ -3+/- i * sqrt( 11 ) ] / 2
And that's about as far as we're going to get for x
using the quadratic equation.
And we can just solve for x now,
because it...
Sorry.
We have x = [ -3 + i sqrt( 11 ) ] / 2
That's one.
And the next root is going to be
[ -3 - i sqrt( 11 ) ] / 2
And those those are our answers.