Quadratic equations with complex roots

1003102501

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

1003102502

Level: 
A
Find the complex roots of the following quadratic equation. \[ 4x^2 + 0.0025 = 0 \]
\( x_1=-\frac1{40}\mathrm{i}\text{, } x_2=\frac1{40}\mathrm{i} \)
\( x_1=-\frac1{4}\mathrm{i}\text{, } x_2=\frac1{4}\mathrm{i} \)
\( x_1=-\frac5{16}\mathrm{i}\text{, } x_2=\frac5{16}\mathrm{i} \)
\( x_1=-\frac5{4}\mathrm{i}\text{, } x_2=\frac5{4}\mathrm{i} \)

1003102503

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

2000001506

Level: 
A
Factorize the equation \(4x^2+25=0\) in the set of complex numbers.
\( 4\left( x-\frac{5}{2}i\right)\left( x+\frac{5}{2}i\right)=0\)
\(( 2x+5)( 2x+5)=0\)
\( 4\left( x+\frac{5}{2}i\right)\left( x+\frac{5}{2}i\right)=0\)
\( 4\left( x-\frac{5}{2}i\right)\left( x-\frac{5}{2}i\right)=0\)

2000001512

Level: 
A
Let \( x_1=2-\frac{\sqrt{5}}{2}i\) be one of the roots of a quadratic equation with real coefficients. Find the other root \(x_2\) of this equation.
\( x_2 =2+\frac{\sqrt{5}}{2}i\)
\( x_2 =-2-\frac{\sqrt{5}}{2}i\)
\( x_2 =-2+\frac{\sqrt{5}}{2}i\)
\( x_2 = \frac{1}{2-\frac{\sqrt{5}}{2}i}\)