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9000028409

Level:

Find the condition which is equivalent to the fact that the equation

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

with

x \in R

and real coefficients

a

b

c

does not have a real solution.

(b^{2} - 4 a c < 0 and a \neq 0) or (a = b = 0 and c \neq 0)

b^{2} - 4 a c < 0

b^{2} - 4 a c < 0 and a \neq 0

(b^{2} - 4 a c < 0 and a \neq 0) or (a b = 0 and c \neq 0)

9000028408

Level:

Find the condition which is equivalent to the fact that the equation

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

with

x \in R

and real coefficients

a

b

c

has two real solutions and one of the solutions is bigger than the other one.

b^{2} - 4 a c > 0 and a \neq 0

b^{2} - 4 a c \neq 0 and a \neq 0

- \frac{b}{2 a} > \frac{\sqrt{b^{2} - 4 a c}}{2 a}

- \frac{b}{2 a} < \frac{\sqrt{b^{2} - 4 a c}}{2 a}

9000028407

Level:

Find the condition which is equivalent to the fact that the equation

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

with

x \in R

and real coefficients

a

b

c

has a unique positive and a unique negative real solution.

b^{2} - 4 a c > 0 and \frac{c}{a} < 0

b^{2} - 4 a c > 0 and - \frac{b}{2 a} < 0

(\frac{c}{a} < 0) and (\frac{b}{a} > 0)

(\frac{c}{a} < 0) and (\frac{b}{a} < 0)

9000028406

Level:

Find the condition which is equivalent to the fact that the equation

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

with

x \in R

and real coefficients

a

b

c

has a solution in a form of a pair of two opposite real nonzero numbers.

\frac{c}{a} < 0 and b = 0

- \frac{b}{2 a} = 0

b^{2} = 4 a c and a \neq 0

b^{2} = 4 a c and a \neq 0 and c \neq 0

9000028403

Level:

Find the condition which is equivalent to the fact that the equation

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

with

x \in R

and real coefficients

a

b

c

has two real solutions

x_{1} \neq x_{2}

x_{1} > 0

x_{2} > 0

b^{2} - 4 a c > 0 and \frac{c}{a} > 0 and \frac{b}{a} < 0

a \neq 0 and c > 0

a > 0 and b < 0 and c > 0 and b^{2} - 4 a c > 0

a \neq 0 and c > 0 and b^{2} - 4 a c > 0

9000026005

Level:

Which part of the plane describes the solution of the following system of inequalities?

\begin{aligned} x \leq & 3 \\ 5 x > & 9 - 3 y \end{aligned}

yellow

red

green

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9000026006

Level:

In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?

\begin{aligned} x + y & \geq 3 \\ y - 2 x & < - 1 \end{aligned}

\begin{aligned} x + y & > 3 \\ y - 2 x & < - 1 \end{aligned}

\begin{aligned} x + y & \leq 3 \\ y - 2 x & < - 1 \end{aligned}

\begin{aligned} x + y & < 3 \\ y - 2 x & > - 1 \end{aligned}

9000026007

Level:

In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?

\begin{aligned} y & < 2 \\ y + 1 & \geq x + 1 \end{aligned}

\begin{aligned} y & \geq 2 \\ y + 1 & < x + 1 \end{aligned}

\begin{aligned} y & > 2 \\ y + 1 & \leq x + 1 \end{aligned}

\begin{aligned} y & \leq 2 \\ y & > x \end{aligned}

9000026008

Level:

In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?

\begin{aligned} 2 x - y & \leq 2 \\ 2 x + y & \geq - 2 \end{aligned}

\begin{aligned} 2 x - y & \geq 2 \\ 2 x + y & \geq - 2 \end{aligned}

\begin{aligned} 2 x - y & \leq 2 \\ 2 x + y & \leq - 2 \end{aligned}

\begin{aligned} 2 x - y & \geq 2 \\ 2 x + y & \leq - 2 \end{aligned}

9000026009

Level:

In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?

\begin{aligned} 2 y - x & < 4 \\ x - 2 y & < 2 \end{aligned}

\begin{aligned} 2 y - x & < 4 \\ x - 2 y & > 2 \end{aligned}

\begin{aligned} 2 y - x & > 4 \\ 2 y - x & < - 2 \end{aligned}

\begin{aligned} 2 y - x & > 4 \\ 2 y - x & > - 2 \end{aligned}

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