Quadratic Discriminant

Quadratic Discriminant: Level 4 Challenges


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


\(k\) 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 \(b^2-4ac.\)


Provided that two roots of the above equation are real and distinct for \(a \in \mathbb{R}-A,\) find the set \(A.\)


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