 par jarmtak
Publié: 17 juin 2022 (il y a 2 semaines)
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Baixar Gta Iv Super Compactado 875mb 2012golkes

Baixar Gta Iv Super Compactado 875mb 2012golkes Â . a API Â .Q:

A complex number satisfies \$|z^2 + z + 1| = |z^3 + z^2 + z + 1|\$. Find \$z\$.

I need some help with this problem. The first part of the problem seemed simple: I used the Heron formula to find the minimum and maximum. So we have:
\$sqrt{left(frac{z + 1}{z}right)^2 – 1} leq |z| leq sqrt{left(frac{z + 1}{z}right)^2 + 1}\$.
It follows that \$|z| leq 1\$, and I plugged these bounds into the original problem, found the minimum, and then plugged that into the quadratic equation. I got the roots \$z = frac{1 pm sqrt{5}}{2}\$, and by plugging the second root into the original expression, I got the final answer of \$z = frac{sqrt{5} + 1}{2}\$. So I’m 99% sure I got the solution. However, the homework does not specify that I should use the Heron formula or otherwise find the minimum or maximum. I know it’s guaranteed to have a real solution, but I’m just not sure of how to find the correct solution. Thanks!

A:

For a complex number \$z\$, the magnitude \$|z|\$ is defined as \$sqrt{x^2 + y^2}\$. Using this, we have
\$\$
|z^2 + z + 1| = sqrt{(z^2 + z + 1)^2} = sqrt{z^4 + 2z^3 + 3z^2 + z + 1} = sqrt{(z^2)^2 + 2z^2 + 1} cdot sqrt{3} = sqrt{3}|z|.
\$\$
Now consider \$|z^3 + z^2 + z + 1|\$. We have
begin{align}
|z^3 + z^2 + z + 1| &= sqrt{(z^3 + z^2 + z + 1)^2} = sq