1003067802
Level:
C
For \( x\in (-\infty;3) \) choose the correct form of the equation
\[ \left|x^2- 9 x + 20\right|=1 \]
that does not contain an absolute value.
\( x^2-9x+20=1 \)
\( -x^2+9x-20=1 \)
\( x^2-9x+20=-1 \)
\( x^2+9x-20=1 \)
Quadratic equations and inequalities
1003047001
Level:
A
We are given the equation \( 2x^2+10x=8x+2x^2 \). Decide which of the following equations has the different set of roots than the given equation has, i.e. choose the equation which is not equivalent to the given equation.
\( 2x+10=8+2x \)
\( 10x=8x \)
\( 2x^2+2x=2x^2 \)
\( x^2+5x=4x+x^2 \)
1003020001
Level:
A
Find the sum of the roots of the equation.
\[
4x^2+2x=0
\]
\(-\frac12\)
\(0\)
\(2\)
\(-2\)
9000039004
Level:
B
Find all \(x\) such that the expression \( (x - 1)(x - 7) \) is negative.
\(x\in (1;7)\)
\(x\in (-\infty ;1)\cup (7;+\infty )\)
No such \(x\)
exists.
\(x\in \mathbb{R}\)
9000034910
Level:
B
Find the solution set of the following quadratic inequality.
\[
(x - 3)^{2}\geq 0
\]
\(\mathbb{R}\)
\(\emptyset \)
\(\mathbb{R}\setminus \{3\}\)
\(\{3\}\)
9000034906
Level:
B
The solution set of one of the following quadratic inequalities is
\(\left (-\infty ;-\frac{3}
{5}\right )\cup \left (\frac{1}
{6};\infty \right )\). Determine this inequality.
\(\left (5x + 3\right )\left (1 - 6x\right ) < 0\)
\(\left (5x - 3\right )\left (6x + 1\right ) < 0\)
\(\left (5x + 3\right )\left (1 - 6x\right ) > 0\)
\(\left (5x - 3\right )\left (6x + 1\right ) > 0\)
9000034907
Level:
B
Find all \(x\in \mathbb{R}\)
for which the following expression takes nonnegative values.
\[
-2\left (x - 3\right )\left (2 - x\right )
\]
\(\left (-\infty ;2\right ] \cup \left [ 3;\infty \right )\)
\(\left [ 2;3\right ] \)
\(\left (2;3\right )\)
\(\left (-\infty ;2\right )\cup \left (3;\infty \right )\)
9000034908
Level:
B
Find all \(x\in \mathbb{R}\)
for which the following expression is nonpositive.
\[
\left (x + 1\right )\left (4 + x\right )
\]
\(\left [ -4;-1\right ] \)
\(\left (-\infty ;-4\right ] \cup \left [ -1;\infty \right )\)
\(\left (-4;-1\right )\)
\(\left (-\infty ;-4\right )\cup \left (-1;\infty \right )\)
9000034909
Level:
B
Find the solution set of the following quadratic inequality.
\[
-3(x + 2)^{2} < 0
\]
\(\mathbb{R}\setminus \{ - 2\}\)
\(\emptyset \)
\(\{- 2\}\)
\(\mathbb{R}\)
9000034905
Level:
B
The solution set of one of the following quadratic inequalities is the interval
\(\left [ -\frac{7}
{6}; \frac{3}
{4}\right ] \).
Determine this inequality.
\(\left (x + \frac{7}
{6}\right )\left (x -\frac{3}
{4}\right )\leq 0\)
\(\left (x + \frac{7}
{6}\right )\left (x -\frac{3}
{4}\right )\geq 0\)
\(\left (x -\frac{7}
{6}\right )\left (x + \frac{3}
{4}\right )\geq 0\)
\(\left (x -\frac{7}
{6}\right )\left (x + \frac{3}
{4}\right )\leq 0\)