Kvadratické rovnice v obore komplex. čísel

9000035602

Časť: 
C
Nájdite hodnoty parametra \(m\in \mathbb{C}\), pre ktoré má daná kvadratická rovnica dvojnásobný koreň. \[ mx^{2} - 2x - 1 + \mathrm{i} = 0 \]
\(m = -\frac{1} {2} -\frac{1} {2}\mathrm{i}\)
\(m = -1\)
\(m = -1 + \mathrm{i}\)
\(m = -\frac{1} {2} + \frac{1} {2}\mathrm{i}\)

9000035608

Časť: 
C
Rovnica \[ x^{2} - 2\mathrm{i}x + q = 0 \] s parametrom \(q\in \mathbb{C}\) má jeden koreň \(x_{1} = 1 + 2\mathrm{i}\). Nájdite druhý koreň \(x_{2}\) a parameter \(q\).
\(x_{2} = -1,\ q = -1 - 2\mathrm{i}\)
\(x_{2} = -1 - 4\mathrm{i},\ q = 9 - 6\mathrm{i}\)
\(x_{2} = 1 - 4\mathrm{i},\ q = 7 - 4\mathrm{i}\)
\(x_{2} = 1,\ q = -1 - 2\mathrm{i}\)
\(x_{2} = -1,\ q = 1 + 2\mathrm{i}\)

9000035609

Časť: 
C
Rovnica \[ x^{2} + px - 11 = 0 \] s parametrom \(p\in \mathbb{C}\) má jeden koreň \(x_{1} = 3 -\mathrm{i}\sqrt{2}\). Nájdite druhý koreň \(x_{2}\) a parameter \(p\).
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 2\mathrm{i}\sqrt{2}\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\sqrt{2}\)