Solve the quadratic equation $z^{2} - (8 + 3 i) z + 13 = - 13 i$ .

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Solution

Consider the given equation.

$z^{2} - (8 + 3 i) z + 13 = - 13 i$

$z^{2} - (8 + 3 i) z + 13 (1 + i) = 0$

We know that

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Therefore,

$z = \frac{8 + 3 i \pm \sqrt{{(8 + 3 i)}^{2} - 4 \times 1 \times 13 (1 + i)}}{2}$

$z = \frac{8 + 3 i \pm \sqrt{64 + 9 i^{2} + 48 i - 52 - 52 i}}{2}$

$z = \frac{8 + 3 i \pm \sqrt{64 - 9 - 52 - 4 i}}{2}$

$z = \frac{8 + 3 i \pm \sqrt{3 - 4 i}}{2}$

Hence, the value of $z$ is $\frac{8 + 3 i \pm \sqrt{3 - 4 i}}{2}$ .

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