query: The probability that the equation will have real roots is
positive_passages.docid: 155791
positive_passages.text: Probability that $ax^2+bx+c$ has no real roots after rolling 3 dice.
negative_passages.docid: 128998
negative_passages.text: Relation between real roots of a polynomial and real roots of its derivative
negative_passages.docid: 63689
negative_passages.text: proving roots of equation $x^2+x+4f=0$ are real.
negative_passages.docid: 87331
negative_passages.text: Real polynomial in two variables
negative_passages.docid: 45811
negative_passages.text: How to prove that r is a double root if and only if it is a root of a polynomial and of its derivative.
negative_passages.docid: 11664
negative_passages.text: If $a_0+\frac{a_1}{2}+\dots + \frac{a_n}{n+1}=0$ then $a_nx^n+\dots + a_0$ has a root in $[0,1]$
negative_passages.docid: 54093
negative_passages.text: Probability of Obtaining the Roots in a Quadratic Equation by Throwing a Die Three Times
negative_passages.docid: 158063
negative_passages.text: A formula for the roots of a solvable polynomial
negative_passages.docid: 156842
negative_passages.text: Random quadratic equation
negative_passages.docid: 69288
negative_passages.text: Number of real roots of $\sum_{k=0}^{n}\frac{x^{k}}{k!}=0$
negative_passages.docid: 18477
negative_passages.text: $(W_1+W_2+\cdots+W_n)^a \leq W_1^a +\cdots + W_n^a$ for $n$ integer, $n\geq 2$, $W\gt 0$ and $a$ constant, real, $0\lt a\lt 1$
negative_passages.docid: 76175
negative_passages.text: Does the equation have a real solution...please help!
negative_passages.docid: 112709
negative_passages.text: $\frac{1}{x-a} + \frac{1}{x-b} + \frac{1}{x-c} = 0 $ has precisely two real roots
negative_passages.docid: 56697
negative_passages.text: How to show that the given equation has at least one complex root ,a s.t |a|>12
negative_passages.docid: 141093
negative_passages.text: How many real roots does $(x-a)^3+(x-b)^3+(x-c)^3$ have?
negative_passages.docid: 51804
negative_passages.text: How to show that a root of the equation $x (x+1)(x+2) ....... (x+2009) = c $ can have multiplicity at most 2?
negative_passages.docid: 22192
negative_passages.text: System of linear equations having a real solution has also a rational solution.