What is a quadratic equation and how is it solved?

A A second grade equationalso known as quadratic equationis an equation with the standard form axtwo + bx + c = 0, a second degree polynomial. It consists of a single variable, xand three terms, a, b and cso that the sum of a by the square of xplus b by xand more cis equal to zero, and being to non-zero.

Quadratic Equation Formula

The quadratic equation was already used by the mathematicians of Babylon 2000 years BC, although the first complete solution did not arrive until the 9th century from the hand of the mathematician Al-Khuarismi. With it, numerous physics problems involving parabolic phenomena have been solved.

Various methods can be applied to find the value of the unknown. One of the most used is the method of complete the squaresfrom which the following formula is derived to solve the equation:

Solution of a quadratic equation

Note that ± is used, since the equation can have one or two solutions. Knowing the value of the group b2–4ac (what remains under the square root), we can already know the type of result that we will obtain. This group is called discriminating and is represented by the Greek letter Δ (delta).


Depending on the value of the discriminant, three different types of results can be obtained.

1.- Δ > 0 (delta greater than zero): If delta is positive, then its square root has two possible solutions, one positive and one negative. This is reflected in the graph of the equation: a parabola that cuts the abscissa axis at two points (at two values ​​of x).

2.- Δ = 0 (delta equal to zero): There is only one possible solution. In this case, the parabola intersects the x-axis only once.

Solution if delta equals zero

3.- Δ < 0 (delta less than zero): The parabola does not cut the abscissa axis; the solution is not a real number. There are two solutions with complex conjugate roots in which appears the imaginary number i.

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