Can the discriminant be used to analyze the behavior of solutions near caustics in optics or wave propagation?

Simplify. Check the solutions. Using the Discriminant, b2 − 4ac, to Determine the Number and Type of Solutions of a Quadratic Equation. The discriminant of a quadratic is the expression inside the radical of the quadratic formula. b2−4(ac) b 2 - 4 ( a ... Recall from precalcu- lus algebra that if the discriminant b2 - 4ac is positive, the equation (2) has two distinct ... 2 ) + b(c1y0. 1 + c2y0. 2) + c(c1y1 + ... (2ax + b)2 = b2 − 4ac. ⇒. 2ax + b = ±. √ b2 − 4ac. ⇒. 2ax = −b ±. √ b2 − 4ac. ⇒ x = −b±. √ b2−4ac. 2a . This is the quadratic formula. 1Jason Zimba is a ... Jan 24, 2019 ... ... 2+bx+c=0 if and only if. x={-b± √{b2-4ac}}/{2a}. LaTeX Original \documentclass{article} \usepackage{amssymb, mathrsfs, enumitem} \usepackage ... 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 ... If b2 – 4ac < 0, then the graph is an ellipse (if b = 0 and a = c, the graph is a circle). Let's investigate it! Open the Quadratic Equations applet: In the ... Jun 3, 2016 ... Let λ± = −b±. √ b2−4ac. 2a be the roots of p (λ) and suppose for the moment that b2 − 4ac 6= 0. From Eq. (1.3) it follows that for any ... b = (2siso + ses) (Ved – Ves). c = ses (ses + siso) (Ves – Ved). Taking into account that smax > siso the answer of the sum is. x = {-b - √ (b2 – 4ac)} / 2a. Sep 3, 2020 ... Explanation. In the quadratic formula, −b±√(b2−4ac) / 2a, the term b2−4ac is called the discriminant. The discriminant plays an important role ...