How is the discriminant related to the theory of splines and piecewise polynomial functions?

Jun 20, 2014 ... ... (b2-4ac ≥ 0) ⇒ (x = {-b-√{b2-4ac}}/{2a}) \lor (x = {-b+√{b2-4ac}}/{2a}). Please write complete proofs in each subquestion, without ... We can compute the discriminant of any power of a polynomial. For example, the quadratic discriminant is given by Δ 2 = b 2 − 4 a c \Delta_2 = b^2 - 4ac ... Oct 20, 2011 ... (a) Prove that the discriminant D = b2 - 4ac ∈ K has a square root. /. D in L and that we have L = K(. /. D). [Hint: use the quadratic formula ... 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 ... May 30, 2023 ... The square root of the discriminant gives us (Edit: double?) the distance that each quadratic root lies either side of where the axis of symmetry crosses the x ... The discriminant (B2-4AC) is used to determine which conic section will result. If the discriminant is less than zero we have a circle (if A = C) or an ellipse; The discriminant is B^2-4AC. Notice this is the portion of the formula inside the square root. If the discriminant = 0, then the formula degrades to x = -B ... The discriminant of a quadratic is the expression inside the radical of the quadratic formula. b2−4(ac) b 2 - 4 ( a ... Feb 22, 2011 ... ... b2– 4ac < 0. We are now able to compute the coefficient of ... if n is sufficiently large, max (a, b) << c, c > 0 and b2– 4ac < 0 ... 2 )2 - b2. 4 + c. Quadratic formula: the roots of ax2 + bx + c are -b 소. /b2 - 4ac. 2a . Exponent rules: ab · ac = ab+c ab ac = ab−c. (ab)c = abc a1/b = b.