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# Complex Numbers

"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.

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 \(x?\)

**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 \(i^2 = -1\).

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