Quadratic formula

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Statement

Consider a quadratic equation of the form:

ax^2 + bx + c = 0, a,b,c \in \R, a \ne 0

This is treated as an equation in x. The quadratic formula is a formula to determine the solutions of this equation. The short version of the formula is that the roots are:

\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

if the expression makes sens; otherwise there are no roots.

To understand the formula, first define the discriminant of the quadratic function ax^2 + bx + c as the value b^2 - 4ac. Now, we make three cases:

Case for discriminant b^2 - 4ac Conclusion for roots Conclusion for factorization of polynomial ax^2 + bx + c
positive, i.e., b^2 - 4ac > 0 there are two real roots, given as \frac{-b + \sqrt{b^2 - 4ac}}{2a} and \frac{-b - \sqrt{b^2 - 4ac}}{2a} If we denote the roots by \alpha, \beta, then ax^2 + bx + c = a(x- \alpha)(x - \beta)
zero, i.e., b^2 - 4ac = 0 there is a single real root with multiplicity two, and that root is -b/2a ax^2 + bx + c = a(x + (b/2a))^2
negative, i.e., b^2 - 4ac < 0 there are no real roots the polynomial does not factor, i.e., it is irreducible