Algebra
# 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 $f(x) = ax^2 + bx + c$.

$k$ 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 $\lfloor -3.4 \rfloor = -4$.