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 \(p\).
\(10\)
\(- 10\)
\(3\)
\(- 1\)
Quadratic Equations with Complex Roots
9000069901
Level:
A
Solve the following quadratic equation in the complex plane.
\[
x^{2} + 4x + 5 = 0
\]
\(x_{1} = -2 + \mathrm{i}\),
\(x_{2} = -2 -\mathrm{i}\)
\(x = -2\)
\(x_{1} = 2 + \mathrm{i}\),
\(x_{2} = 2 -\mathrm{i}\)
\(x_{1} = -3\),
\(x_{2} = -1\)
9000069902
Level:
A
Solve the following quadratic equation in the complex plane.
\[
3x^{2} + 2x + 2 = 0
\]
\(x_{1} = -\frac{1}
{3} + \frac{\sqrt{5}}
{3} \mathrm{i}\),
\(x_{2} = -\frac{1}
{3} -\frac{\sqrt{5}}
{3} \mathrm{i}\)
\(x_{1} = -\frac{1}
{3}\)
\(x_{1} = \frac{1}
{3} + \frac{\sqrt{5}}
{3} \),
\(x_{2} = \frac{1}
{3} + \frac{\sqrt{5}}
{3} \)
\(x_{1} = \frac{1}
{3} + \frac{\sqrt{5}}
{3} \mathrm{i}\),
\(x_{2} = \frac{1}
{3} -\frac{\sqrt{5}}
{3} \mathrm{i}\)
9000069903
Level:
A
Find the factorization of the quadratic polynomial
\[
x^{2} + 2x + 2
\]
in the set of polynomials with complex valued coefficients.
\((x + 1 + \mathrm{i})(x + 1 -\mathrm{i})\)
\((x - 1 + \mathrm{i})(x - 1 -\mathrm{i})\)
\((x -\mathrm{i})(x + \mathrm{i})\)
\((x - 1 + \mathrm{i})(x + 1 -\mathrm{i})\)
9000069904
Level:
A
Find the factorization of the quadratic polynomial
\[
x^{2} + 2x + 5
\]
in the set of polynomials with complex valued coefficients.
\((x + 1 - 2\mathrm{i})(x + 1 + 2\mathrm{i})\)
\((x - 1 - 2\mathrm{i})(x - 1 + 2\mathrm{i})\)
\((x + 1 - 2\mathrm{i})(x - 1 + 2\mathrm{i})\)
\((x - 1 - 2\mathrm{i})(x + 1 + 2\mathrm{i})\)
9000069905
Level:
A
Find the sum of all the complex solutions of the following quadratic equation.
\[
5x^{2} + 4x + 8 = 0
\]
\(-\frac{4}
{5}\)
\(- \frac{4}
{10}\)
\(\frac{24}
{5} \mathrm{i}\)
\(0\)
9000064505
Level:
A
Find the factorization of the following quadratic polynomial in the set of polynomial
with complex valued coefficients.
\[
2x^{2} + 32
\]
\(2(x + 4\mathrm{i})(x - 4\mathrm{i})\)
\(2(x - 4\mathrm{i})^{2}\)
\((x + 4\mathrm{i})(x - 4\mathrm{i})\)
\(2(x + 4\mathrm{i})^{2}\)
9000064507
Level:
A
Solve the following quadratic equation in the complex plane.
\[
4x^{2} + 12 = 0
\]
\(x_{1, 2} =\pm \mathrm{i}\sqrt{3}\)
\(x_{1, 2} =\pm 3\)
\(x_{1, 2} =\pm 3\mathrm{i}\)
\(x_{1, 2} =\pm \sqrt{3}\)
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.
\[
2x^{2} + 4x + 5
\]
\(2\! \left (x + 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x + 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(2\! \left (x - 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x - 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(\left (x + 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x + 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(\left (x - 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x - 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\)