Cuadráticas

Cuadráticas refers to quadratic equations, which are polynomial equations of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'x' is the variable. The coefficient 'a' cannot be zero; otherwise, the equation would be linear.

The graph of a quadratic equation is a parabola, a symmetrical U-shaped curve. The direction of the parabola depends on the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, indicating a minimum value. If 'a' is negative, the parabola opens downwards, indicating a maximum value.

The solutions to a quadratic equation, also known as roots, represent the points where the parabola intersects the x-axis. These roots can be found using various methods, including factoring, completing the square, or the quadratic formula. The quadratic formula is given by x = [-b ± sqrt(b^2 - 4ac)] / 2a.

The discriminant, which is the part of the quadratic formula under the square root (b^2 - 4ac), determines the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If the discriminant is negative, there are no real roots, but two complex conjugate roots.

Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and mathematics. They are used to model projectile motion, optimize profit, analyze curves, and solve a wide range of problems involving second-degree relationships. Understanding cuadráticas is fundamental for further studies in algebra and calculus.