For z^2 − 2z + 5 = 0, the roots are z = 1 ± 2i. What are |z| and the arguments Arg(z) for these roots?

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Multiple Choice

For z^2 − 2z + 5 = 0, the roots are z = 1 ± 2i. What are |z| and the arguments Arg(z) for these roots?

The important idea is that a complex number z = x + yi has a modulus |z| = sqrt(x^2 + y^2) and an argument Arg(z) = the angle θ with tan θ = y/x, taking the correct quadrant. For z = 1 + 2i, |z| = sqrt(1^2 + 2^2) = sqrt(5). Arg(z) = arctan(2/1) = arctan(2) ≈ 63.435°, since it lies in the first quadrant. For z = 1 − 2i, |z| is still sqrt(5). Arg(z) = arctan((-2)/1) = −arctan(2) ≈ −63.435°, since it lies in the fourth quadrant. So the roots share |z| = sqrt(5), and their arguments are approximately ±63.435°. The angle values 0° or 45° don’t fit because tan θ would have to be 0 or 1, not 2.

The important idea is that a complex number z = x + yi has a modulus |z| = sqrt(x^2 + y^2) and an argument Arg(z) = the angle θ with tan θ = y/x, taking the correct quadrant.

For z = 1 + 2i, |z| = sqrt(1^2 + 2^2) = sqrt(5). Arg(z) = arctan(2/1) = arctan(2) ≈ 63.435°, since it lies in the first quadrant.

For z = 1 − 2i, |z| is still sqrt(5). Arg(z) = arctan((-2)/1) = −arctan(2) ≈ −63.435°, since it lies in the fourth quadrant.

So the roots share |z| = sqrt(5), and their arguments are approximately ±63.435°. The angle values 0° or 45° don’t fit because tan θ would have to be 0 or 1, not 2.

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