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# Complex numbers question

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1. (i) Find the roots of the equation z^2 + (2 sqrt 3)z +4 = 0, giving your answers in the form x + iy, where x and y are real.
(ii) State the modulus and argument of each root.
(iii) Showing all your working, verify that each root also satisfies the equation
z^6 = −64.

How do I do the third question without expanding the roots of z to the power 6?
2. (Original post by bmqib)
(i) Find the roots of the equation z^2 + (2 sqrt 3)z +4 = 0, giving your answers in the form x + iy, where x and y are real.
(ii) State the modulus and argument of each root.
(iii) Showing all your working, verify that each root also satisfies the equation
z^6 = −64.

How do I do the third question without expanding the roots of z to the power 6?
Do you know that Mod(z^6)=(modz)^6 and that arg(z^6) = 6x(arg z)?
(or put another way, De Moivre's theorem)

3. Assuming OP doesn't know De Moivre. Split the quartic into the product of two quadratics - you know what one of them is going to be.
4. (Original post by tiny hobbit)
Do you know that Mod(z^6)=(modz)^6 and that arg(z^6) = 6x(arg z)?
(or put another way, De Moivre's theorem)
Yeah but that's not in the curriculum so there could be a different way to do it... do I have to find z in the r(sin theta + i cos theta) form then?
5. (Original post by Mr M)

Assuming OP doesn't know De Moivre. Solve the disguised quadratic.
But how do I factorise it that quickly? and the right hand side is not equal to lhs?
6. (Original post by bmqib)
But how do I factorise it that quickly? and the right hand side is not equal to lhs?
It is now I corrected the wrong sign!

http://www.purplemath.com/modules/specfact2.htm
7. (Original post by bmqib)
(i) Find the roots of the equation z^2 + (2 sqrt 3)z +4 = 0, giving your answers in the form x + iy, where x and y are real.
(ii) State the modulus and argument of each root.
(iii) Showing all your working, verify that each root also satisfies the equation
z^6 = −64.

How do I do the third question without expanding the roots of z to the power 6?
Lol, I just did the same paper myself. Wanted to post the same question. CIE sucks.

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