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I need to solve this question:

$$ z^2 - 2(4+i)z+10- 20i = 0 $$

Can you help me please , thank you

KReiser
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Iss Hak
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  • this question and answers can be helpful https://math.stackexchange.com/questions/3042441/find-all-the-solutions-of-z2-13iz-8-i-0/3042458#3042458 – user376343 Jun 24 '20 at 08:05

1 Answers1

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Hint:

The solutions to the equation $az^2 + bz + c = 0$ can be given by $$z = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$$

This is the quadratic formula.

In this case, $a = 1$, $b = -2(4+ i)$ and $c=10-20i$, so substitute these values and do the algebra.

The tricky part is evaluating $b^2 - 4ac$, but just remember how to expand brackets and make sure you use $i^2 = -1$.

Edit:

Note you probably do not have to simplify the radial and it will likely be acceptable for you to leave your solutions in the form $z = p \pm \sqrt{q + ri}$

Alt:

If you don't like the quadratic formula, you can try to complete the square. Here would be the first step: $$[x-(4 + i)]^2 - (4 + i)^2 + 10 - 20i = 0$$ Again, expand the brackets and then re-arrange.

The lesson is: solve this quadratic with complex coefficients in the way you would usually solve a quadratic equation using whatever your favourite approach is.

PhysicsMathsLove
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    Do you think the op will be able to simplify the radicand? – fleablood Jun 23 '20 at 22:42
  • Edited with some more guidance. Given the question, they probably won't have to find the square root of the discriminant - it's nasty. In terms of the complex algebra, they probably have some experience so best not to super explicitly show it and let them figure it out – PhysicsMathsLove Jun 23 '20 at 22:54