Complex numbers question

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Complex numbers question

Postby umricky » Sun Nov 12, 2023 11:45 am

If z=1+2i is a root of the equation z^2 = az+ b find the values of a and b

I have no idea how to answer this, but I'm sure it's pretty simple.

Thanks in advance!
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Re: Complex numbers question

Postby Guest » Mon Nov 13, 2023 7:31 am

We will first find the complex conjugate of z.

Given that z = 1 + 2i, the complex conjugate of z, denoted as z*, is 1 - 2i.

Now, we'll use the property that if z is a root, its conjugate z* is also a root. Therefore, we can write the equation as:

[tex](1+2i)^2 = a(1+2i) + b[/tex] and [tex](1-2i)^2 = a(1-2i) + b[/tex]

Solving these two simultaneous equations:

Hence, [tex]a = -2 + 2i[/tex]
[tex]b = 3 + 8i[/tex]

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