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I have a hard time solving the following equation.

$$z^{2} + (1 - i) \cdot z - i = 0$$

I tried factorization and got $z = -1$ or $z = i$, which I suppose are correct.

However, when I try the quadratic formula for solving second degree equations, the results I get don't match with the solutions I get above.

amWhy
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Chrysa
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  • Do you mean that you tried the quadratic formula? – Toby Mak Jan 01 '18 at 23:07
  • Well, you simply made computation mistakes with the quadratic formula, because it actually returns you $-1$ and $i$. – Crostul Jan 01 '18 at 23:08
  • Thank you. I will check it again. – Chrysa Jan 01 '18 at 23:10
  • $z^{2} + (1 - i) \cdot z - i = 0.;$ Note that $a = 1, ; b = 1-i,; c =- i$. Use the quadradic formula, $$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \iff z= \frac{(i-1 \pm \sqrt{ (1-i)^2 +4i} }{2}$$ and simplify. – amWhy Jan 01 '18 at 23:37
  • I don't think this question, beyond the first two answers, needs additional answers. Overkill. – amWhy Jan 01 '18 at 23:46

3 Answers3

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$$b^2-4ac = (1-i)^2 - 4(-i) = -2i+4i= 2i = (i+1)^2$$ Hence the two solutions are $$z_1= \frac{-1+i+(i+1)}{2}=i; \ \ \ \ z_2 = \frac{-1+i-(i+1)}{2}=-1$$

Crostul
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  • I see why $$ 2i = (i + 1)^2. $$ Is this a common trick when solving equations with complex numbers? – Chrysa Jan 01 '18 at 23:17
  • @Chrysa This trick is evident when you compute the discriminant of your particular equation. First: you need to evaluate $(1-i)^2=-2i$; second: you need to find some number whose square is $2i$. Well, simply pick $i(1-i)$, i.e. $1+i$. In general, to compute a square root in $\Bbb C$ you have to switch to polar coordinates. – Crostul Jan 02 '18 at 09:40
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Useful facts $$\sqrt{x+yi}=\pm \left( \sqrt{\frac{\sqrt{x^2+y^2}+x}{2}} +\frac{iy}{|y|}\sqrt{\frac{\sqrt{x^2+y^2}-x}{2}} \right)$$

\begin{align} z &= \frac{i-1 \pm \sqrt{(1-i)^2+4i}}{2} \\ &= \frac{i-1 \pm \sqrt{2i}}{2} \\ &= \frac{i-1 \pm (1+i)}{2} \\ &= -1 \quad \text{or} \quad i \end{align}

Refer to another answer here

Ng Chung Tak
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Just sorting for the equations on the real and imaginary coordinates, $$ x^2 - y^2 + x + y = 0 \\ 2xy + y - x - 1 = 0 $$ we get these possible solutions:

enter image description here

The intersection points $(-1, 0) = -1$ and $(0, 1) = i$ give the solutions.

mvw
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