If z1 = 3 + 2i and z1 = 2 − 4i and |z1 + z2|2+|z1 − z2|2is equal

It seems like there's a mistake in your question. You've assigned two different values to z1. Let's assume the first one is z1 and the second one is z2. So, z1 = 3 + 2i and z2 = 2 - 4i.

The expression you want to calculate is |z1 + z2|^2 + |z1 - z2|^2.

Step 1: Calculate z1 + z2 and z1 - z2. z1 + z2 = (3 + 2i) + (2 - 4i) = 5 - 2i z1 - z2 = (3 + 2i) - (2 - 4i) = 1 + 6i

Step 2: Calculate the absolute values (or magnitudes) of these complex numbers. |z1 + z2| = sqrt((5)^2 + (-2)^2) = sqrt(29) |z1 - z2| = sqrt((1)^2 + (6)^2) = sqrt(37)

Step 3: Square these absolute values. |z1 + z2|^2 = (sqrt(29))^2 = 29 |z1 - z2|^2 = (sqrt(37))^2 = 37

Step 4: Add these squared absolute values together. |z1 + z2|^2 + |z1 - z2|^2 = 29 + 37 = 66

So, |z1 + z2|^2 + |z1 - z2|^2 = 66.