Quadratic equations and inequalities

9000020408

Level: 
A
Which of the given equations have at least one root the same? \[ \begin{aligned} x^{2} + 8x + 15 & = 0 &\text{(1)} \\x^{2} - 8x + 15 & = 0 &\text{(2)} \\x^{2} +\phantom{ 8}x - 12 & = 0 &\text{(3)} \\x^{2} - 2x -\phantom{ 1}8 & = 0 &\text{(4)} \\\end{aligned}\]
equations (2) and (3)
equations (1) and (3)
equations (2) and (4)
Such a pair does not exist.

9000021803

Level: 
B
Solve the following inequality. \[ (3x - 1)(2 - 4x) < 0 \]
\(x\in \left (-\infty ; \frac{1} {3}\right )\cup \left (\frac{1} {2};\infty \right )\)
\(x\in \left (\frac{1} {3}; \frac{1} {2}\right )\)
\(x\in \left (-\infty ; \frac{1} {2}\right )\)
\(x\in \left (\frac{1} {3};\infty \right )\)

9000020409

Level: 
B
One of the solutions of the quadratic equation \( x^{2} + bx - 10 = 0\) is \(x_{1} = 5\). Find the second solution \(x_{2}\) and the value of the coefficient \(b\).
\(x_{2} = -2\) and \(b = -3\)
\(x_{2} = -3\) and \(b = -2\)
\(x_{2} = 2\) and \(b = 3\)
\(x_{2} = 3\) and \(b = 2\)