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This one number can tell you whether the solutions to a quadratic equation are real or non-real, and whether they are distinct or repeated.

# Quadratic Discriminant: Level 4 Challenges

What is the sum of all the integers $$a$$ such that the following equation has no real roots:

$\frac{x^2+x+a-5}{x-1}=a?$

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

Find the sum of integral values of $$\alpha$$ such that $$2\log (x+3)=\log (\alpha x)$$ has exactly one real solution and $$|\alpha |<20$$.

If $$\alpha$$ is one of the non-real seventh roots of unity, then find the discriminant of the monic quadratic equation with the roots $$\alpha+\alpha^2+\alpha^4$$ and $$\alpha^3+\alpha^5+\alpha^6$$.

 Details and assumptions
The discriminant of a quadratic equation $$ax^2+bx+c=0$$ is $$b^2-4ac.$$

$(a-1)(x^{2}+x+1)^{2}-(a+1)(x^{4}+x^{2}+1)=0$ Provided that two roots of the above equation are real and distinct for $$a \in \mathbb{R}-A$$. then find the set $$A$$.

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