Complex Variables and Partial Differential Equations (3130005)

Question-1b

BE | Semester-3 Summer-2020 | 27-10-2020

Q1) (b)

Find the roots of the equation, $z^{2} - (5 + i) z + (8 + i) = 0$ .

z^{2} - (5 + i) z + (8 + i) = 0

\Rightarrow a = 1; b = - (5 + i); c = (8 + i)

z = \frac{- b \pm \sqrt{b^{2} - 4 ac}}{2 a}

\Rightarrow z = \frac{(5 + i) \pm \sqrt{{(5 + i)}^{2} - 4 (1) (8 + i)}}{2 (1)}

\Rightarrow z = \frac{(5 + i) \pm \sqrt{25 + 10 i - 1 - 32 - 4 i}}{2}

\Rightarrow z = \frac{(5 + i) \pm \sqrt{- 8 + 6 i}}{2}

\Rightarrow z = \frac{(5 + i) \pm (1 + 3 i)}{2} (∵ {(1 + 3 i)}^{2} = 1 + 6 i - 9 = - 8 + 6 i)

\Rightarrow z = \frac{(5 + i) + (1 + 3 i)}{2} OR \frac{(5 + i) - (1 + 3 i)}{2}

\Rightarrow z = \frac{6 + 4 i}{2} OR \frac{4 - 2 i}{2}

\Rightarrow z = 3 + 2 i OR 2 - i

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