ruutvõrrandid

Ruutvõrrandid, also known as quadratic equations, are algebraic 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, as this would reduce the equation to a linear one.

The solutions to a quadratic equation are called roots. A quadratic equation can have zero, one, or two distinct real roots. The nature and number of roots are determined by the discriminant, which is calculated as delta = b^2 - 4ac. If delta is positive, there are two distinct real roots. If delta is zero, there is exactly one real root (a repeated root). If delta is negative, there are no real roots, but there are two complex conjugate roots.

There are several methods to solve quadratic equations. The most common include factoring, completing the square, and using the quadratic formula. The quadratic formula, x = (-b ± sqrt(b^2 - 4ac)) / 2a, directly provides the roots of the equation, regardless of the nature of the discriminant. Factoring is often the quickest method when applicable, but not all quadratic equations can be easily factored. Completing the square is a systematic method that can be used to derive the quadratic formula itself and is useful for understanding the structure of quadratic expressions.

Quadratic equations have wide-ranging applications in various fields, including physics, engineering, economics, and geometry. They are used to model projectile motion, calculate areas, determine optimal values in business, and solve problems involving curves and parabolas.