Quadratic Equations with Complex Roots

9000064506

Level: 
A
Find the factorization of the following quadratic polynomial in the set of polynomial with complex valued coefficients. \[ 2x^{2} + 4x + 5 \]
\(2\! \left (x + 1 + \frac{\sqrt{6}} {2} \mathrm{i}\right )\! \! \left (x + 1 -\frac{\sqrt{6}} {2} \mathrm{i}\right )\)
\(2\! \left (x - 1 + \frac{\sqrt{6}} {2} \mathrm{i}\right )\! \! \left (x - 1 -\frac{\sqrt{6}} {2} \mathrm{i}\right )\)
\(\left (x + 1 -\frac{\sqrt{6}} {2} \mathrm{i}\right )\! \! \left (x + 1 + \frac{\sqrt{6}} {2} \mathrm{i}\right )\)
\(\left (x - 1 -\frac{\sqrt{6}} {2} \mathrm{i}\right )\! \! \left (x - 1 + \frac{\sqrt{6}} {2} \mathrm{i}\right )\)

9000039106

Level: 
B
Find the value of the parameter \(a\) which guarantees that the quadratic equation \[ x^{2} + 2ax + a = 0 \] has a pair of complex conjugate solutions with a nonzero imaginary part.
\(a\in (0,1)\)
\(a\in [ 0,1] \)
\(a\in (-\infty ,0)\cup (1,\infty )\)
Such an \(a\) does not exist

9000035602

Level: 
C
Find the values of the parameter \(m\in \mathbb{C}\) which guarantee that the following quadratic equation has a double solution. \[ mx^{2} - 2x - 1 + \mathrm{i} = 0 \]
\(m = -\frac{1} {2} -\frac{1} {2}\mathrm{i}\)
\(m = -1\)
\(m = -1 + \mathrm{i}\)
\(m = -\frac{1} {2} + \frac{1} {2}\mathrm{i}\)

9000035605

Level: 
B
The number \(\cos \frac{7} {6}\pi + \mathrm{i}\sin \frac{7} {6}\pi \) is a solution of a quadratic equation with real valued coefficients. Find the second solution.
\(\cos \frac{5} {6}\pi + \mathrm{i}\sin \frac{5} {6}\pi \)
\(\cos \frac{1} {6}\pi + \mathrm{i}\sin \frac{1} {6}\pi \)
\(\cos \frac{7} {6}\pi + \mathrm{i}\sin \frac{7} {6}\pi \)
\(\cos \frac{11} {6} \pi + \mathrm{i}\sin \frac{11} {6} \pi \)

9000035609

Level: 
C
One of the roots of the equation \( x^{2} + px - 11 = 0\) with the parameter \(p\in \mathbb{C}\) is \(x_{1} = 3 -\mathrm{i}\sqrt{2}\). Find the second root \(x_{2}\) and the corresponding value of the parameter \(p\).
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 2\mathrm{i}\sqrt{2}\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\sqrt{2}\)