Question #40464

The question tests the ability to compute the square root of a complex discriminant and identify extraneous roots introduced by modulus equations.

The trap lies in the condition $|z| = 5(z + \bar{z}) - 35$.

Students often square such equations to form $|z|^2 = (10\text{Re}(z) - 35)^2$, which destroys the non-negativity constraint of the modulus function ($10\text{Re}(z) - 35 \ge 0$).

Consequently, the extraneous root $z = 3+4i$ perfectly satisfies the squared equation ($25 = 25$) but fails the original equation ($5 \neq -5$), leading to an incorrect final calculation if not verified.