Quadratic Equations Question - None

Question

Solve the equation $(1 + i)x^2 + (1 - i)x - 2i = 0$ and find $|\alpha - \beta|^2$ where $\alpha, \beta$ are the roots.

Accepted Answer

Step 1: Divide the equation by $(1+i)$:

$$x^2 + \frac{(1-i)}{(1+i)}x - \frac{2i}{(1+i)} = 0$$

Step 2: Simplify to get $x^2 - ix - (1+i) = 0$

Step 3: By Vieta's formulas: $\alpha + \beta = i$ and $\alpha\beta = -(1+i)$

Step 4: Calculate $\alpha - \beta = \sqrt{(\alpha+\beta)^2 - 4\alpha\beta} = \sqrt{i^2 + 4(1+i)} = \sqrt{-1+4+4i} = \sqrt{3+4i}$

Step 5: $|\alpha - \beta| = \sqrt{9+16} = 5$

Step 6: Therefore $|\alpha - \beta|^2 = 5$

∴ Answer is 5.

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