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 \(i^2 = -1\).
Let \(z=x(1-i)+8(-2+i),\) where \(x\) is a real number. If \(z^2\) is a negative real number, what is the value of \(x?\)
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 \(i^2 = -1\).
\( \frac{12 - \sqrt{-9}}{12 + \sqrt{-9}} + \frac{12 + \sqrt{-9}}{12 - \sqrt{-9}} \) can be simplified to \( \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. What is the value of \( a + b \)?