Quadratic equations with complex roots

2010013311

Level:

Find the quadratic equation with real coefficients such that one of the solutions is the complex number

x_{1} = 2 (\cos \frac{11 π}{6} + i \sin \frac{11 π}{6})

x^{2} - 2 \sqrt{3} x + 4 = 0

x^{2} + 2 \sqrt{3} x + 4 = 0

x^{2} + 4 x + 2 \sqrt{3} = 0

x^{2} - 2 x + 4 = 0

2010013310

Level:

Find the quadratic equation with real coefficients such that one of the solutions is the complex number

x_{1} = 2 (\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})

x^{2} + 2 x + 4 = 0

x^{2} - 2 x + 4 = 0

x^{2} + 4 x + 2 = 0

x^{2} + 2 \sqrt{3} x + 4 = 0

2010013309

Level:

Find the complex roots of the following quadratic equation.

x^{2} - (2 + 2 i) x + 2 i = 0

x_{1, 2} = 1 + i

x_{1} = 1 + i, x_{2} = 1 - i

x_{1, 2} = - 1 - i

x_{1} = 1 + i, x_{2} = - 1 - i

2010013308

Level:

Find the solution set of the following quadratic equation in the complex plane.

2 x^{2} - (2 - 4 i) x + 3 - 2 i = 0

{\frac{1}{2} + \frac{1}{2} i; \frac{1}{2} - \frac{5}{2} i}

{\frac{1}{2} + \frac{1}{2} i; \frac{1}{2} - \frac{1}{2} i}

\emptyset

{- \frac{1}{2} - \frac{1}{2} i; - \frac{1}{2} + \frac{5}{2} i}

{1 + i; 1 - 5 i}

2010013307

Level:

Find the values of the real coefficients

a

b

and

c

such that the quadratic equation

a x^{2} + b x + c = 0

has solutions

x_{1, 2} = \frac{1}{2} \pm i

a = 4, b = - 4, c = 5

a = 4, b = 4, c = 5

a = 5, b = - 5, c = 4

a = - 4, b = 4, c = 5

2010013306

Level:

Find the set of all values of the parameter

p \in R

for which the following quadratic equation has solutions with nonzero imaginary part.

9 p x^{2} + 5 x + p = 0

(- \infty; - \frac{5}{6}) \cup (\frac{5}{6}; \infty)

(- \frac{5}{6}; \frac{5}{6})

(\frac{5}{6}; \infty)

{- \frac{5}{6}; \frac{5}{6}}

R ∖ {- \frac{5}{6}; \frac{5}{6}}

2010013305

Level:

The number

\sqrt{2} (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})

is a solution of a quadratic equation with real valued coefficients. Find the second solution.

\sqrt{2} (\cos \frac{5 π}{4} + i \sin \frac{5 π}{4})

\sqrt{2} (\cos \frac{π}{4} + i \sin \frac{π}{4})

\sqrt{2} (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})

\sqrt{2} (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})

2010013304

Level:

Find the values of the parameter

p \in R

which guarantee that the equation

x^{2} - 2 p x + 4 = 0

has solution with a nonzero imaginary part.

p \in (- 2; 2)

p \in (- \infty; - 2)

p \in (2; \infty)

p \in \emptyset

2010013303

Level:

Find the quadratic equation with the solution

x_{1, 2} = \pm 3 i

x^{2} + 9 = 0

x^{2} - 9 i = 0

x^{2} - 9 = 0

x^{2} + 9 i = 0

2010013302

Level:

Find the quadratic equation with real coefficients such that one of the solutions is the complex number

x_{1} = 1 - 2 \sqrt{2} i

x^{2} - 2 x + 9 = 0

x^{2} + 2 x - 9 = 0

x^{2} - 2 x - 7 = 0

x^{2} + 9 x - 2 = 0

Quadratic equations with complex roots

2010013311

2010013310

2010013309

2010013308

2010013307

2010013306

2010013305

2010013304

2010013303

2010013302

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