Editing Quadratic equation (section)

===Geometric interpretation===
{{quadratic_function_graph_complex_roots.svg}}
The function {{math|''f''(''x'') {{=}} ''ax''<sup>2</sup> + ''bx'' + ''c''}} is a [[quadratic function]].<ref>{{cite book |last=Wharton |first=P. |title=Essentials of Edexcel Gcse Math/Higher |year=2006 |publisher=Lonsdale |isbn=978-1-905-129-78-2|url=https://books.google.com/books?id=LMmKq-feEUoC&q=%22Quadratic+function%22+%22Quadratic+equation%22&pg=PA63 |page=63}}</ref> The graph of any quadratic function has the same general shape, which is called a [[parabola]]. The location and size of the parabola, and how it opens, depend on the values of {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}. If {{math|''a'' &gt; 0}}, the parabola has a minimum point and opens upward. If {{math|''a'' &lt; 0}}, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its [[vertex (curve)|vertex]]. The ''{{math|x}}-coordinate'' of the vertex will be located at <math>\scriptstyle x=\tfrac{-b}{2a}</math>, and the ''{{math|y}}-coordinate'' of the vertex may be found by substituting this ''{{math|x}}-value'' into the function. The ''{{math|y}}-intercept'' is located at the point {{math|(0, ''c'')}}.

The solutions of the quadratic equation {{math|''ax''<sup>2</sup> + {{math|''bx''}} + {{math|''c''}} {{=}} 0}} correspond to the [[root of a function|roots]] of the function {{math|''f''(''x'') {{=}} ''ax''<sup>2</sup> + ''bx'' + ''c''}}, since they are the values of {{math|''x''}} for which {{math|''f''(''x'') {{=}} 0}}. If {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are [[real numbers]] and the [[domain of a function|domain]] of {{math|''f''}} is the set of real numbers, then the roots of {{math|''f''}} are exactly the {{math|''x''}}-[[coordinates]] of the points where the graph touches the {{math|''x''}}-axis. If the discriminant is positive, the graph touches the [[x-axis|{{math|''x''}}-axis]] at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the {{math|''x''}}-axis.