Question

The sum of the squares of the imaginary roots of the equation $ z^8 - 20z^4 + 64 = 0 $ is:

• A. $0$
• B. $-12$
• C. $-4$
• D. $-16$

Hint:

When solving equations like $ z^n = a $, use substitution to reduce the degree of the equation. Identify roots using polar form or known identities, and isolate the required type of roots (e.g., real or imaginary) before computing expressions like sum of squares.

Answer 1

The Correct Option is B

Step 1: Analyze the given equation.
We are given: $$ z^8 - 20z^4 + 64 = 0. $$ Let \( w = z^4 \). Then the equation becomes: $$ w^2 - 20w + 64 = 0. $$ Solve using the quadratic formula: $$ w = \frac{20 \pm \sqrt{(-20)^2 - 4(1)(64)}}{2} = \frac{20 \pm \sqrt{400 - 256}}{2} = \frac{20 \pm \sqrt{144}}{2} = \frac{20 \pm 12}{2}. $$ So: $$ w = 16 \quad \text{or} \quad w = 4. $$ Thus: $$ z^4 = 16 \quad \text{and} \quad z^4 = 4. $$ Step 2: Find the roots of \( z^4 = 16 \).
The fourth roots of 16 are: $$ z = 2, \quad z = 2i, \quad z = -2, \quad z = -2i. $$ Imaginary roots: \( z = 2i, -2i \)
Step 3: Find the roots of \( z^4 = 4 \).
The fourth roots of 4 are: $$ z = \sqrt{2}, \quad z = i\sqrt{2}, \quad z = -\sqrt{2}, \quad z = -i\sqrt{2}. $$ Imaginary roots: \( z = i\sqrt{2}, -i\sqrt{2} \)
Step 4: List all imaginary roots.
Imaginary roots are: $$ z = 2i, -2i, i\sqrt{2}, -i\sqrt{2} $$ Step 5: Compute the sum of their squares.
$$ (2i)^2 + (-2i)^2 + (i\sqrt{2})^2 + (-i\sqrt{2})^2 = -4 - 4 - 2 - 2 = -12. $$ Step 6: Final Answer.
$$ \boxed{-12} $$