Editing Quadratic equation (section)

===Discriminant===
[[File:Quadratic eq discriminant.svg|thumb|right|Figure 3. Discriminant signs|alt=Figure 3. This figure plots three quadratic functions on a single Cartesian plane graph to illustrate the effects of discriminant values. When the discriminant, delta, is positive, the parabola intersects the {{math|''x''}}-axis at two points. When delta is zero, the vertex of the parabola touches the {{math|''x''}}-axis at a single point. When delta is negative, the parabola does not intersect the {{math|''x''}}-axis at all.]]
In the quadratic formula, the expression underneath the square root sign is called the ''[[discriminant]]'' of the quadratic equation, and is often represented using an upper case {{math|''D''}} or an upper case Greek [[Delta (letter)|delta]]:<ref>'''Δ''' is the initial of the [[Greek language|Greek]] word '''Δ'''ιακρίνουσα, ''Diakrínousa'', discriminant.</ref>
<math display="block">\Delta = b^2 - 4ac.</math>
A quadratic equation with ''real'' coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

*If the discriminant is positive, then there are two distinct roots <math display="block">\frac{-b + \sqrt {\Delta}}{2a} \quad\text{and}\quad \frac{-b - \sqrt {\Delta}}{2a},</math> both of which are real numbers. For quadratic equations with [[rational number|rational]] coefficients, if the discriminant is a [[square number]], then the roots are rational—in other cases they may be [[quadratic irrational]]s.
*If the discriminant is zero, then there is exactly one [[real number|real]] root <math>-\frac{b}{2a},</math> sometimes called a repeated or [[multiple root|double root]] or two equal roots.
*If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) [[complex number|complex]] roots<ref>{{cite book|last1=Achatz|first1=Thomas|last2=Anderson|first2=John G.|last3=McKenzie|first3=Kathleen|title=Technical Shop Mathematics|year=2005|publisher=Industrial Press|isbn=978-0-8311-3086-2|url=https://books.google.com/books?id=YOdtemSmzQQC&q=quadratic+formula&pg=PA276|page=277}}</ref><math display="block">
-\frac{b}{2a} + i \frac{\sqrt {-\Delta}}{2a} \quad\text{and}\quad -\frac{b}{2a} - i \frac{\sqrt {-\Delta}}{2a},
</math> which are [[complex conjugate]]s of each other. In these expressions {{math|''i''}} is the [[imaginary unit]].

Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.