Quadratic Discriminant
What is the sum of all the integers \(a\) such that the following equation has no real roots:
\[\frac{x^2+x+a-5}{x-1}=a?\]
by
Sandeep Mehra
\(k\) 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 \(\lfloor -3.4 \rfloor = -4\).
by
Calvin Lin
Find the sum of integral values of \(\alpha\) such that \(2\log (x+3)=\log (\alpha x)\) has exactly one real solution and \(|\alpha |<20\).
by
Pranjal Jain
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 \(b^2-4ac.\)
by
Sandeep Bhardwaj
\[(a-1)(x^{2}+x+1)^{2}-(a+1)(x^{4}+x^{2}+1)=0\]
Provided that two roots of the above equation are real and distinct for \(a \in \mathbb{R}-A,\) find the set \(A.\)
by
Sandeep Bhardwaj