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

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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 \(i^2 = -1\).

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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 \(i^2 = -1\).

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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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