Question

If \( 2z = 7 + i\sqrt{3} \), then the value of \( z^2 - 7z + 4 \) is:

• A. \( \frac{39}{4} \)
• B. \( \frac{39}{4} \)
• C. -9
• D. 17
• E. 9

Hint:

Be sure to correctly distribute and simplify terms when working with complex numbers and their algebraic expressions to avoid errors in calculations.

Answer 1

The Correct Option is C

First, solve for \( z \) from the given equation: \[ z = \frac{7 + i\sqrt{3}}{2} \] Calculate \( z^2 \): \[ z^2 = \left( \frac{7 + i\sqrt{3}}{2} \right)^2 = \frac{49 + 14i\sqrt{3} - 3}{4} = \frac{46 + 14i\sqrt{3}}{4} = \frac{23}{2} + \frac{7i\sqrt{3}}{2} \] Calculate \( 7z \): \[ 7z = 7 \times \frac{7 + i\sqrt{3}}{2} = \frac{49}{2} + \frac{7i\sqrt{3}}{2} \] Substitute \( z^2 \) and \( 7z \) into the expression \( z^2 - 7z + 4 \): \[ z^2 - 7z + 4 = \left(\frac{23}{2} + \frac{7i\sqrt{3}}{2}\right) - \left(\frac{49}{2} + \frac{7i\sqrt{3}}{2}\right) + 4 \] \[ = \frac{23}{2} + \frac{7i\sqrt{3}}{2} - \frac{49}{2} - \frac{7i\sqrt{3}}{2} + 4 \] \[ = \frac{23 - 49 + 8}{2} \] \[ = \frac{-18}{2} \] \[ = -9 \] Therefore, the value of \( z^2 - 7z + 4 \) is \(-9\).