Quadratic Equations with Complex Roots

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 \(p\).
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 2\mathrm{i}\sqrt{2}\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\sqrt{2}\)

9000035601

Level: 
B
Find the values of the parameter \(p\in \mathbb{R}\) which guarantee that the following quadratic equation has solutions with nonzero imaginary part. \[ px^{2} - 3x + 4p = 0 \]
\(p\in \left (-\infty ,-\frac{3} {4}\right )\cup \left (\frac{3} {4},\infty \right )\)
\(p\in\left (-\frac{3} {4}, \frac{3} {4}\right )\)
\(p\in\left (\frac{3} {4},\infty \right )\)
\(p\in\left \{-\frac{3} {4}, \frac{3} {4}\right \}\)
\(p\in\mathbb{R}\setminus \left \{-\frac{3} {4}, \frac{3} {4}\right \}\)

9000035603

Level: 
A
Find the solution set of the following equation. \[ 4x^{2} + 9 = 0 \]
\(\left \{-\frac{3} {2}\mathrm{i}, \frac{3} {2}\mathrm{i}\right \}\)
\(\left \{-\frac{2} {3}\mathrm{i}, \frac{2} {3}\mathrm{i}\right \}\)
\(\left \{-\frac{9} {4}\mathrm{i}, \frac{9} {4}\mathrm{i}\right \}\)
\(\left \{-\frac{3} {2}, \frac{3} {2}\right \}\)

9000022803

Level: 
B
Establish the values of the parameter \(t\) which ensure that the equation \[ x^{2} + tx + t + 8 = 0 \] with an unknown \(x\) has complex solutions with a nonzero imaginary part.
\(\left (-4,8\right )\)
\(\left [ -4,8\right ] \)
\(\left (-\infty ,-4\right )\cup \left (8,\infty \right )\)
\(\left (-\infty ,-4\right ] \cup \left [ 8,\infty \right )\)

9000019808

Level: 
B
Assuming \(x\in \mathbb{C}\), find the solution set of the following equation. \[ x\left (x + 1\right )\left (x^{2} + 1\right ) = 0 \]
\(\left \{-1,0,-\mathrm{i},\mathrm{i}\right \}\)
\(\left \{-1,0,1,-\mathrm{i},\mathrm{i}\right \}\)
\(\left \{-1,1,-\mathrm{i},\mathrm{i}\right \}\)
\(\left \{-1,0,-\mathrm{i}\right \}\)