Neliöyhtälön

Neliöyhtälö, often referred to as a quadratic equation in English, is a polynomial equation of the second degree. This means it is an equation containing one variable, say x, that can be written in the standard form ax^2 + bx + c = 0, where a, b, and c are coefficients (constants) and the coefficient 'a' is not equal to zero. The highest power of the variable in the equation is two, hence the term "neliö" (square).

The solutions to a quadratic equation are called roots or zeros. A quadratic equation can have zero, one, or two distinct real roots. The nature and number of roots can be determined by the discriminant, which is the part of the quadratic formula under the square root sign: b^2 - 4ac. 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; instead, there are two complex conjugate roots.

The most common method for solving a quadratic equation is using the quadratic formula, which provides the values of x that satisfy the equation: x = [-b ± sqrt(b^2 - 4ac)] / 2a. Other methods include factoring, completing the square, and graphical methods. Factoring is applicable when the quadratic expression can be easily broken down into two linear factors. Completing the square is a systematic approach that can be used to derive the quadratic formula itself. Graphically, the roots of a quadratic equation correspond to the x-intercepts of the parabola represented by the function y = ax^2 + bx + c.