Quadratic Equations with Complex Roots

1003107507

Level: 
C
Find the solution set of the following quadratic equation in the complex plane. \[ x^2 +3\mathrm{i}x + 10 = 0 \]
\( \{-5\mathrm{i}; 2\mathrm{i}\} \)
\( \{-2\mathrm{i}; 5\mathrm{i}\} \)
\( \{2-5\mathrm{i}; 5-2\mathrm{i}\} \)
\( \{-2+5\mathrm{i}; -5+2\mathrm{i}\} \)
\( \emptyset \)

1003107506

Level: 
C
Find the complex roots of the following quadratic equation. \[ 9x^2 + 4\mathrm{i} = 0 \]
\( x_1=-\frac{\sqrt2}3+\frac{\sqrt2}3\mathrm{i}\text{, }\ x_2=\frac{\sqrt2}3-\frac{\sqrt2}3\mathrm{i} \)
\( x_1=\frac23\mathrm{i}\text{, }\ x_2=-\frac23\mathrm{i} \)
\( x_1=\frac{\sqrt2}3+\frac{\sqrt2}3\mathrm{i}\text{, }\ x_2=-\frac{\sqrt2}3-\frac{\sqrt2}3\mathrm{i} \)
\( x_1=\frac23\text{, }\ x_2=-\frac23 \)

1003107505

Level: 
C
Find the complex roots of the following quadratic equation. \[ 4\mathrm{i}x^2 + 1 = 0 \]
\( x_1=\frac{\sqrt2}4+\frac{\sqrt2}4\mathrm{i}\text{, }\ x_2=-\frac{\sqrt2}4-\frac{\sqrt2}4\mathrm{i} \)
\( x_1=-\frac{\sqrt2}4+\frac{\sqrt2}4\mathrm{i}\text{, }\ x_2=\frac{\sqrt2}4-\frac{\sqrt2}4\mathrm{i} \)
\( x_1=\frac12+\frac12\mathrm{i}\text{, }\ x_2=-\frac12-\frac12\mathrm{i} \)
\( x_1=-\frac12+\frac12\mathrm{i}\text{, }\ x_2=\frac12-\frac12\mathrm{i} \)

1003107504

Level: 
C
Find the complex roots of the following quadratic equation. \[ 16\mathrm{i}x^2- 9\mathrm{i}^3 = 0 \]
\( x_1=\frac34\mathrm{i}\text{, }\ x_2=-\frac34\mathrm{i} \)
\( x_1=\frac34\text{, }\ x_2=-\frac34\)
\( x_1=\frac43\mathrm{i}\text{, }\ x_2=-\frac43\mathrm{i} \)
\( x_1=\frac43\text{, }\ x_2=-\frac43 \)

1003107503

Level: 
C
Find the complex roots of the following quadratic equation. \[ (3-\mathrm{i})x^2-(1-2\mathrm{i})x = 0 \]
\( x_1=0\text{, }\ x_2=\frac12-\frac12\mathrm{i} \)
\( x_1=0\text{, }\ x_2=-\frac12+\frac12\mathrm{i} \)
\( x_1=0\text{, }\ x_2=\frac12+\frac12\mathrm{i} \)
\( x_1=0\text{, }\ x_2=-\frac12-\frac12\mathrm{i} \)

1003107501

Level: 
C
Find the complex roots of the following quadratic equation. \[ 2\mathrm{i}\,x^2 - 5\mathrm{i}\,x = 0 \]
\( x_1=0\text{, }\ x_2 = \frac52 \)
\( x_1=0\text{, }\ x_2 = \frac52\mathrm{i} \)
\( x_1=0\text{, }\ x_2 = -\frac52 \)
\( x_1=0\text{, }\ x_2 = -\frac52\mathrm{i} \)

1003102505

Level: 
B
Find the set of all complex roots of the following equation. \[ \left(x^2- 2x + 5\right)\left(x^2 + 6x + 10\right) = 0 \]
\( \{1\pm2\mathrm{i}; -3\pm\mathrm{i} \} \)
\( \{-1\pm2\mathrm{i}; -3\pm\mathrm{i} \} \)
\( \{-1\pm2\mathrm{i}; 3\pm\mathrm{i} \} \)
\( \{1\pm2\mathrm{i}; 3\pm\mathrm{i} \} \)

1003102504

Level: 
B
Find the set of all complex roots of the following quadratic equation. \[ x^2 + 4x + 8 = 0 \]
\( \left\{ 2\sqrt2\left(\cos\frac{3\pi}4+\mathrm{i}\cdot\sin\frac{3\pi}4\right); 2\sqrt2\left(\cos\frac{5\pi}4+\mathrm{i}\cdot\sin\frac{5\pi}4\right) \right\} \)
\( \left\{ 2\left(\cos\frac{3\pi}4+\mathrm{i}\cdot\sin\frac{3\pi}4\right); 2\left(\cos\frac{5\pi}4+\mathrm{i}\cdot\sin\frac{5\pi}4\right) \right\} \)
\( \left\{ 2\sqrt2\left(\cos\frac{\pi}4+\mathrm{i}\cdot\sin\frac{\pi}4\right); 2\sqrt2\left(\cos\frac{7\pi}4+\mathrm{i}\cdot\sin\frac{7\pi}4\right) \right\} \)
\( \left\{ 2\left(\cos\frac{\pi}4+\mathrm{i}\cdot\sin\frac{\pi}4\right); 2\left(\cos\frac{7\pi}4+\mathrm{i}\cdot\sin\frac{7\pi}4\right) \right\} \)

1003102503

Level: 
A
Find the complex roots of the following quadratic equation. \[ 5x^2 + 12 = 0 \]
\( x_1=-\frac{2\sqrt{15}}5\mathrm{i}\text{, }x_2=\frac{2\sqrt{15}}5\mathrm{i} \)
\( x_1=-\frac{\sqrt{15}}5\mathrm{i}\text{, }x_2=\frac{\sqrt{15}}5\mathrm{i} \)
\( x_1=-\frac{\sqrt{12}}5\mathrm{i}\text{, }x_2=\frac{\sqrt{12}}5\mathrm{i} \)
\( x_1=-\frac{2\sqrt{3}}5\mathrm{i}\text{, }x_2=\frac{2\sqrt{3}}5\mathrm{i} \)