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Question
Solve the following quadratic equation:
\[x^2 - \left( 5 - i \right) x + \left( 18 + i \right) = 0\]
Solution
\[ x^2 - \left( 5 - i \right)x + \left( 18 + i \right) = 0\]
\[\text { Comparing the given equation with the general form } a x^2 + bx + c = 0, \text { we get }\]
\[a = 1, b = - \left( 5 - i \right) \text { and } c = \left( 18 + i \right)\]
\[x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm \sqrt{\left( 5 - i \right)^2 - 4\left( 18 + i \right)}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm \sqrt{\left( 5 - i \right)^2 - 4\left( 18 + i \right)}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm \sqrt{- 48 - 14i}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm i\sqrt{48 + 14i}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm i\sqrt{49 - 1 + 2 \times 7 \times i}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm i\sqrt{\left( 7 + i \right)^2}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm i\left( 7 + i \right)}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) + i\left( 7 + i \right)}{2} \text { or }x = \frac{\left( 5 - i \right) - i\left( 7 + i \right)}{2}\]
\[ \Rightarrow x = 2 + 3i, 3 - 4i\]
\[\text { So, the roots of the given quadratic equation are } 2 + 3i \text { and } 3 - 4i .\]
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