What are Sturm sequences, and how are they related to the discriminant?

Note: The number b2 - 4ac is called the discriminant of the equation ax2 + bx + c = 0. If the discriminant is negative then the equation has no real solutions. Square root in the denominator ; − b {\displaystyle -b}. {\displaystyle -b} ; + b 2 − 4 a c {\displaystyle \textstyle +{\sqrt {b^{2}-4ac}}}. {\displaystyle \ ... May 31, 2018 ... ... (2,Z) is denoted by h(d) and called the class number of binary quadratic forms of discriminant d. Note: d = b2 − 4ac, so d ≡ 0 or 1 mod 4. Oct 2, 2022 ... Find all the roots of the solution, and compute the distance between successive roots to see that it is a constant. The discriminant of a quadratic is the expression inside the radical of the quadratic formula. b2−4(ac) b 2 - 4 ( a ... a x2 + bx +c = 0. has roots. x = {-b ± √{b2 - 4ac}}/{2a}. Newton's Method. If an initial guess x0 is close enough to a root of a function g, then the iteration ... b = 1/2 a. 29. + á

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• Consider a PDE with a characteristic equation; might the discriminant of this equation play a role in determining the type of PDE (e.g., elliptic, parabolic, hyperbolic)?
• How does the discriminant relate to the notion of smoothness of a curve or surface?
• What are caustics?
• What are the stability conditions for linear difference equations, and how might a discriminant be relevant?
• In what contexts is the exact value of the discriminant used for further calculations or analysis?
• What is the sign of a quadratic expression when the discriminant is negative and 'a' is positive?
• How is the discriminant related to the Mordell-Weil theorem for elliptic curves over the rationals?
• What are some applications of quadratic forms and their discriminants in areas like physics (e.g., potential energy)?
• How does the sign of the discriminant of a cubic polynomial relate to the number of real roots?
• How is the discriminant related to the concept of algebraic independence?