What are some applications of discriminants in areas like coding theory involving algebraic structures beyond standard polynomials over fields?

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 ... The discriminant of a quadratic is the expression inside the radical of the quadratic formula. b2−4(ac) b 2 - 4 ( a ... Summary. The symbol over the b2-4ac is called a 'radical', or a square root symbol; ' ± ' means 'plus or minus'; The DISCRIMINANT is the expression under ... ... 2 (x + (b / 2a))2 = -4ac / 4a2 + b2 / 4a2 x + (b / 2a)2 = (b2 - 4ac) / 4a2 x + (b / 2a) = +/- sqrt ((b2 - 4ac) / 4a2) x = -b / 2a +/- sqrt(b2 - 4ac) / 2a 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; 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 ... and adding a constant (b/2a)2 so as to make a perfect square: a2x2 + bx + ... If b2 - 4ac is negative this solution will involve complex numbers, which ... where a, b and c are real coefficients. The general solution to such an equation is given by the following formula, called the quadratic formula;. If b2 -4ac ... b2 – 4ac < 0. No real roots. 4. b2 – 4ac = 32- 4x1x(-2) = 9 +8 = 17. b2 ... If b2 – 4ac < 0 , the line does not touch the curve. It is not a tangent. If ...