If and are the roots , then is equal to : [2025]

Question

If $α + i β$ and $γ + i δ$ are the roots of $x^{2} - (3 - 2 i) x - (2 i - 2) = 0, i = \sqrt{- 1}$ , then $α γ + β δ$ is equal to : [2025]

Smart Achievers · Accepted Answer

(4)

We have, $x^{2} - (3 - 2 i) x - (2 i - 2) = 0$

$x = \frac{(3 - 2 i) \pm \sqrt{{(3 - 2 i)}^{2} + 4 (1) (2 i - 2)}}{2}$

$= \frac{(3 - 2 i) \pm \sqrt{9 - 4 - 12 i + 8 i - 8}}{2} = \frac{(3 - 2 i) \pm \sqrt{- 3 - 4 i}}{2}$

$= \frac{(3 - 2 i) \pm \sqrt{{{(1)}^{2} + (2 i)}^{2} - 2 (1) (2 i)}}{2}$

$= \frac{(3 - 2 i) \pm \sqrt{{(1 - 2 i)}^{2}}}{2} = \frac{(3 - 2 i) \pm (1 - 2 i)}{2}$

$∴ x = \frac{3 - 2 i + 1 - 2 i}{2} or \frac{3 - 2 i - 1 + 2 i}{2}$

$∴ x = 2 - 2 i or 1 + 0 i$

So, $α γ + β δ = 2 (1) + (- 2) (0) = 2$ .

Want Us to Email you About Special Offers & Updates?