Quadratic equations with complex roots

1003102501

Level: 
A
Find the set of all complex roots of the following quadratic equation. \[ 9x^2 + 2 = 0 \]
\( \left\{-\frac{\sqrt2}3\mathrm{i}; \frac{\sqrt2}3\mathrm{i}\right\} \)
\( \left\{-\sqrt{\frac{2}3}\mathrm{i}; \sqrt{\frac23}\mathrm{i}\right\} \)
\( \left\{-\frac23\mathrm{i}; \frac23\mathrm{i}\right\} \)
\( \left\{-\frac29\mathrm{i}; \frac29\mathrm{i}\right\} \)
\( \emptyset \)

1003109404

Level: 
C
One of the roots of the quadratic equation \( x^2 + px + 1 - 3\,\mathrm{i} = 0 \) with a complex parameter \( p \) is \( x_1 = -\mathrm{i} \). Choose the equivalent form of the given equation.
\( (x + \mathrm{i})(x -3 - \mathrm{i}) = 0 \)
\( (x + \mathrm{i})(x - 3 +\mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x- 3-\mathrm{i}) = 0 \)
\( (x +\mathrm{i})(x + 3 + \mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x- 3 + \mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x + 3 +\mathrm{i}) = 0 \)

1003109403

Level: 
C
One of the following equations has the solutions \( x_1=\frac12-\mathrm{i} \) and \( x_2=-\frac12+2\,\mathrm{i} \). Find this equation.
\( 4x^2-4\,\mathrm{i}\,x+7+6\,\mathrm{i}=0 \)
\( 4x^2-4\,\mathrm{i}\,x-9+3\,\mathrm{i}=0 \)
\( 4x^2+4\,\mathrm{i}\,x+7+6\,\mathrm{i}\,=0 \)
\( 4x^2+4\,\mathrm{i}\,x-9+3\,\mathrm{i}=0 \)

1003109402

Level: 
C
Which of the given quadratic equations has the roots \( x_1 = 1 +\mathrm{i} \) and \( x_2 = (1 +\mathrm{i})^2 \)?
\( x^2 - (1 + 3\,\mathrm{i})x - 2 + 2\,\mathrm{i} = 0 \)
\( x^2 - (1 + 3\,\mathrm{i})x - 2 - 2\,\mathrm{i} = 0 \)
\( x^2 - (1 + 3\,\mathrm{i})x + 2 + 2\,\mathrm{i} = 0 \)
\( x^2 - (1 + 3\,\mathrm{i})x + 2 - 2\,\mathrm{i} = 0 \)

1003109401

Level: 
C
Which of the given quadratic equations has the roots \( x_1 = 2 + \mathrm{i} \) and \( x_2 = 1 - 3\,\mathrm{i} \)?
\( x^2 - (3 - 2\,\mathrm{i})x + 5 - 5\,\mathrm{i} = 0 \)
\(x^2 + (3 - 2\,\mathrm{i})x + 5 - 5\,\mathrm{i} = 0 \)
\( x^2 - (3 + 2\,\mathrm{i})x + 5 - 5\,\mathrm{i} = 0 \)
\( x^2 + (3 + 2\,\mathrm{i})x + 5 - 5\,\mathrm{i} = 0 \)

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.
\((x + 1 - 2\mathrm{i})(x + 1 + 2\mathrm{i})\)
\((x - 1 - 2\mathrm{i})(x - 1 + 2\mathrm{i})\)
\((x + 1 - 2\mathrm{i})(x - 1 + 2\mathrm{i})\)
\((x - 1 - 2\mathrm{i})(x + 1 + 2\mathrm{i})\)