9000069906
Level:
A
Find the sum of all the complex solutions of the following quadratic equation.
\[
x^{2} - 8x + 17 = 0
\]
\(8\)
\(4\)
\(4\mathrm{i}\)
\(0\)
Quadratic equations with complex roots
9000069910
Level:
B
Find the values of the parameter \(p\in \mathbb{R}\)
which guarantee that the equation
\[
x^{2} + 2px + 16 = 0
\]
has solution with a nonzero imaginary part.
\(p\in (-4;4)\)
\(p\in (-\infty ;4)\)
\(p\in (4;\infty )\)
\(p\in \emptyset\)
9000069907
Level:
B
Find the quadratic equation with real valued coefficients and one of the solutions
\(x_{1} = -5 + \mathrm{i}\).
\(x^{2} + 10x + 26 = 0\)
\(x^{2} - 10x + 26 = 0\)
\(x^{2} - 10x - 24 = 0\)
\(x^{2} + 10x + 24 = 0\)
9000069908
Level:
B
One of the solutions of the quadratic equation
\[
2x^{2} + px + 5 = 0
\]
with a real parameter \(p\)
is
\[
x_1 = -1 + \frac{\sqrt{6}}
{2} \mathrm{i}.
\]
Find the value of \(p\).
\(4\)
\(- 4\)
\(8\)
\(- 8\)
9000069909
Level:
B
One of the solutions of the quadratic equation
\[
9x^{2} - 6x + p = 0
\]
with a real parameter \(p\)
is
\[
x_1=\frac{1}
{3} + \mathrm{i}.
\]
Find the value of \(p\).
\(10\)
\(- 10\)
\(3\)
\(- 1\)
9000064508
Level:
A
Solve the following quadratic equation in the complex plane.
\[
2x^{2} + x + 1 = 0
\]
\(x_{1, 2} = \frac{-1\pm \mathrm{i}\sqrt{7}}
{4} \)
\(x_{1, 2} = \frac{-1\pm \mathrm{i}\sqrt{7}}
{2} \)
\(x_{1, 2} = \frac{1\pm \mathrm{i}\sqrt{7}}
{4} \)
\(x_{1, 2} = \frac{1\pm \mathrm{i}\sqrt{7}}
{2} \)
9000064506
Level:
A
Find the factorization of the following quadratic polynomial in the set of polynomial
with complex valued coefficients.
\[
2x^{2} + 4x + 5
\]
\(2\! \left (x + 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x + 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(2\! \left (x - 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x - 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(\left (x + 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x + 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(\left (x - 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x - 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
9000064501
Level:
B
Find the quadratic equation with the solution
\(x_{1, 2} =\pm 2\mathrm{i}\).
\(x^{2} + 4 = 0\)
\(x^{2} - 4\mathrm{i} = 0\)
\(x^{2} - 4 = 0\)
\(x^{2} + 4\mathrm{i} = 0\)
9000064502
Level:
B
Find the quadratic equation with the solution
\(x_{1, 2} = 2\pm \mathrm{i}\sqrt{2}\).
\(x^{2} - 4x + 6 = 0\)
\(3x^{2} + 4x + 2 = 0\)
\(3x^{2} - 4x + 2 = 0\)
\(x^{2} + 4x + 6 = 0\)
9000064503
Level:
B
Find the values of the real coefficients
\(a\),
\(b\) and
\(c\) such
that the quadratic equation
\[
ax^{2} + bx + c = 0
\]
has solution \(x_{1, 2} =\pm \mathrm{i}\frac{\sqrt{5}}
{3} \).
\(a = 9\text{, }b = 0\text{, }c = 5\)
\(a = 5\text{, }b = 0\text{, }c = 9\)
\(a = 9\text{, }b = 0\text{, }c = -5\)
\(a = 5\text{, }b = 0\text{, }c = -9\)