Find the value of $z_1^2 +z_2^2$?

Asked May 16 '21 at 17:54

Active May 16 '21 at 17:59

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$z_1,z_2$ belongs to $\Bbb C$ (Complex Numbers)

We are given that $z_1(z_1^2-3z_2^2) = 2$ and $z_2(3z_1^2-z_2^2)=1$

If $z_1^2+z_2^2$ belongs to real numbers, then find the value of $z_1^2+z_2^2$?

I tried to use the algebraic identities or taking $z_1 z_2$ to right hand side and subtracting the equations to get the $z_1^2 + z_2^2$ term but to no avail.

Is there a better approach to solve this question?

edited May 16 '21 at 17:59

Bernard

175,478

asked May 16 '21 at 17:54

InfiniteCool23

6

Multiply the second equation by $\pm i$ and add it to the first equation. You get that $(z_1\pm iz_2)^3=1\pm i$. Then multiply those two equation. to get that $(z_1^2+z_2^2)^3=2$. Since $z_1^2+z_2^2$ is real, it must be $\sqrt[3]{2}$. – plop May 16 '21 at 18:05
@plop Thanks for your comment. This approach is really great. – InfiniteCool23 May 16 '21 at 18:13
But I think there is a typo in “1 ± i“, it should be “2 ±i”. – InfiniteCool23 May 16 '21 at 18:15
And thus answer 5^1/3 – InfiniteCool23 May 16 '21 at 18:18
@plop But anyway thanks for your help! – InfiniteCool23 May 16 '21 at 18:23
Yes. I misread it as both expression being equal to $1$. – plop May 16 '21 at 19:45

Find the value of $z_1^2 +z_2^2$?

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