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 \(\lfloor -3.4 \rfloor = -4\).

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