Algebra

# Quadratic Discriminant - Problem Solving

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?

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