Algebra
# Complex Numbers

Let $z = a^2 - 5a - 36 + (a^2 + 6a + 8)i$. What is the sum of all real numbers $a$ such that $z^2$ is a negative real number?

**Details and assumptions**

$i$ is the imaginary number that satisfies $i^2 = -1$.

If the quadratic equation $(2 + i)x^2 + (a - i)x + 46 - 2i = 0$ has real roots, what is the value of the positive real constant $a$?

**Details and assumptions**

$i$ is the imaginary number that satisfies $i^2 = -1$.

Real numbers $x$ and $y$ satisfy the equation: $\frac{x}{1+i} + \frac{y}{1-i} = \frac{148}{12 + 2i}.$ What is the value of $xy?$

**Details and assumptions**

$i$ is the imaginary number satisfying $i^2 = -1$.