Let's go through each problem step by step.
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Simplify \((4 - 12i) - (-8 + 4i)\)
\[ = (4 - 12i) + (8 - 4i) \quad \text{(distributing the negative sign)} \] \[ = (4 + 8) + (-12i - 4i) \] \[ = 12 - 16i \]
The simplified expression is \(12 - 16i\).
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Simplify \((5 + 2i)(1 + 3i)\)
Using the distributive property (FOIL): \[ = 5 \cdot 1 + 5 \cdot 3i + 2i \cdot 1 + 2i \cdot 3i \] \[ = 5 + 15i + 2i + 6i^2 \] Since \(i^2 = -1\): \[ = 5 + 17i + 6(-1) \] \[ = 5 + 17i - 6 \] \[ = -1 + 17i \]
The simplified expression is \(-1 + 17i\).
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Find the total current in the circuit given \(3 - 2j\) amps and \(2 + 4y\) amps.
Add the two currents: \[ (3 - 2j) + (2 + 4y) = (3 + 2) + (-2j + 4y) \] \[ = 5 + (4y - 2j) \]
The total current in the circuit is \(5 + (4y - 2j)\).
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Simplify \(\frac{4 - 2i}{7 + 3i}\)
To simplify, multiply the numerator and the denominator by the conjugate of the denominator: \[ = \frac{(4 - 2i)(7 - 3i)}{(7 + 3i)(7 - 3i)} \] First, compute the denominator: \[ (7 + 3i)(7 - 3i) = 49 + 21i - 21i - 9i^2 = 49 + 9 = 58 \] Now compute the numerator: \[ (4 - 2i)(7 - 3i) = 28 - 12i - 14i + 6i^2 = 28 - 26i + 6(-1) \] \[ = 28 - 26i - 6 = 22 - 26i \] Thus, we have: \[ \frac{22 - 26i}{58} = \frac{22}{58} - \frac{26}{58}i \] Simplifying further: \[ = \frac{11}{29} - \frac{13}{29}i \]
The simplified expression is \(\frac{11}{29} - \frac{13}{29}i\).
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Solve \(5x^2 + 100 = 0\)
First, isolate \(x^2\): \[ 5x^2 = -100 \] \[ x^2 = -20 \] Now take the square root of both sides. This gives: \[ x = \pm \sqrt{-20} = \pm \sqrt{20}i = \pm 2\sqrt{5}i \]
The solutions are \(x = 2\sqrt{5}i\) and \(x = -2\sqrt{5}i\).
In summary:
- \(12 - 16i\)
- \(-1 + 17i\)
- \(5 + (4y - 2j)\)
- \(\frac{11}{29} - \frac{13}{29}i\)
- \(x = 2\sqrt{5}i\) and \(x = -2\sqrt{5}i\)