Can the discriminant be used to study the integrability of dynamical systems?

There will be at most two real solutions to any quadratic equation. The value of b2 − 4ac (known as the discriminant) tells us how many real solutions there are ... Let gcd(a, b, c) = 1, D = b 2 - 4ac. Then as remarked in Mathews ([4, pages 132--133]), then there exist integers #, # such that a# 2 + b## + c# 2 = a ... 1. Let ƒ(x,y) = ax2 + bxy + cy2 be an indefinite binary quadratic form, with real coefficients a, b, c, and discriminant d = b2 —4ac. Then, the discriminant A = b2-4ac satisfies: △ < a². Proof. If the ... + b 2 − a) . In conclusion, we believe that Lemma 1 can be a nice starting. If b2−4ac < 0, change format to linear combination of real-valued functions instead of complex valued func- tions by using Euler's formula. general solution is ... The solution depends on the value of b2 - 4ac. This term is called the "discriminant". There are three cases: If (b2 - 4ac) > 0.0, two real roots exist (i.e ... 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 ... Jun 15, 2023 ... If the discriminant D < 0, then the quadratic equation ax 2 + bx + c = 0 will not have any real roots. Jun 4, 2020 ... I wish I could use b^2-4ac>0 to check whether the people are real or not. If b2 - 4ac is positive (>0) then we have 2 solutions. If b2 - 4ac is 0 then we have only one solution as the formula is reduced to x = [-b ± 0]/ ...