How can the discriminant be used to analyze the intersection of algebraic curves?

The discriminant of a quadratic is the expression inside the radical of the quadratic formula. b2−4(ac) b 2 - 4 ( a ... The discriminant, b2 - 4ac, offers valuable information about the "nature" of the roots of a quadratic equation where a, b and c are rational values. It quickly ... If b2−4ac > 0 the equation has two distinct real number roots. Example. Study some of these examples: Find the roots of x2 + d=b2-4ac No solutions found Rearrange: Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : d-(b^2-4*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. The quadratic formula, x=- b±sqrt(b^2-4ac)/2a, can be derived by completing the square on the general standard form a quadratic equation. 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 ... Feb 2, 2020 ... Overview of the discriminant from quadratic formula and its interpretation. Click here to see all quadratic formula videos: ... The discriminant is 0, so the root is rational and the equation can be solved by factoring. In. , substitute: Since. , the equation has 2 real roots. b2 4ac>0. Jun 4, 2014 ... 3 Answers 3 ... Δ=b2−4ac>0 means that the equation has two real solutions. Δ=b2−4ac<0 means that the equation has no real solutions, but two ...