Quadratic Discriminant
If \(a, b, c \) are real numbers such that \( c (a+b+ c ) < 0 \), what can we say about \( b^2 - 4ac \)?
Hint: Think about the function \( f(x) = ax^2 + bx + c \).
For what values of \(m\) are there no intersection points between the line \( y= mx + 13 \) and the circle \( x^2 + y^2 = 17 \)?
\(k\) is uniformly chosen from the interval \([ -5, 5] \). Let \(p\) be the probability that the quadratic \( f(x) = x^2 + kx + 1 \) has both roots between -2 and 4. What is the value of \( \lfloor 1000 p \rfloor \)?
Details and assumptions
Greatest Integer Function: \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer less than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -3.4 \rfloor = -4\).
For how many positive integer values of c does the equation \( 2x^2 + 705x +c = 0 \) have an integer solution?
For a positive integer \(m\) such that \(m \le 25,\) what is the probability that the quadratic equation \(x^2+mx+\frac{1+m}{2}=0\) has real roots?