Calculating Sum of Squared Magnitudes of Complex Roots
The problem requires finding the value of $|z_1|^2 + |z_2|^2$, where $z_1$ and $z_2$ are the distinct complex solutions to the quadratic equation $z^2 + 4z - (1 + 12i) = 0$. The most direct method is to find the roots.
Step 1: Identify Equation Coefficients
For the quadratic equation $az^2 + bz + c = 0$, we have:
- $a = 1$
- $b = 4$
- $c = -(1 + 12i)$
Step 2: Calculate the Discriminant
The discriminant $\Delta$ is calculated using $\Delta = b^2 - 4ac$.
$ \Delta = 4^2 - 4(1)(-(1 + 12i)) $
$ \Delta = 16 + 4(1 + 12i) $
$ \Delta = 16 + 4 + 48i $
$ \Delta = 20 + 48i $
Step 3: Find the Square Root of the Discriminant
We need to find $\sqrt{20 + 48i}$. Let $\sqrt{20 + 48i} = x + yi$, where $x, y \in \mathbb{R}$.
Squaring gives $(x + yi)^2 = x^2 - y^2 + 2xyi = 20 + 48i$. Equating real and imaginary parts yields:
- $x^2 - y^2 = 20$ (Eq. 1)
- $2xy = 48 \implies xy = 24$ (Eq. 2)
We also use $|x+yi|^2 = |20+48i|$, which gives $x^2 + y^2 = \sqrt{20^2 + 48^2} = \sqrt{400 + 2304} = \sqrt{2704} = 52$ (Eq. 3).
Adding Eq. 1 and Eq. 3: $2x^2 = 72 \implies x^2 = 36 \implies x = \pm 6$.
Subtracting Eq. 1 from Eq. 3: $2y^2 = 32 \implies y^2 = 16 \implies y = \pm 4$.
Since $xy = 24$ (positive), $x$ and $y$ must have the same sign. Therefore, $\sqrt{20 + 48i} = \pm (6 + 4i)$.
Step 4: Find the Complex Roots ($z_1, z_2$)
Using the quadratic formula $z = \frac{-b \pm \sqrt{\Delta}}{2a}$:
$ z = \frac{-4 \pm (6 + 4i)}{2} $
The distinct roots are:
- $z_1 = \frac{-4 + (6 + 4i)}{2} = \frac{2 + 4i}{2} = 1 + 2i$
- $z_2 = \frac{-4 - (6 + 4i)}{2} = \frac{-10 - 4i}{2} = -5 - 2i$
Step 5: Calculate the Squared Magnitudes
The squared magnitude $|z|^2$ for $z = x + yi$ is $x^2 + y^2$.
$ |z_1|^2 = |1 + 2i|^2 = 1^2 + 2^2 = 1 + 4 = 5 $
$ |z_2|^2 = |-5 - 2i|^2 = (-5)^2 + (-2)^2 = 25 + 4 = 29 $
Step 6: Compute the Sum of Squared Magnitudes
The required sum is $|z_1|^2 + |z_2|^2$.
$ |z_1|^2 + |z_2|^2 = 5 + 29 = 34 $
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