Editing Quadratic equation (section)

===Graphical solution===
[[File:Graphical calculation of root of quadratic equation.png|240px|thumb|Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation {{math|2''x''<sup>2</sup> + 4''x'' &minus; 4 {{=}} 0}}. Although the display shows only five significant figures of accuracy, the retrieved value of {{math|''xc''}} is 0.732050807569, accurate to twelve significant figures.]]
[[Image:Visual.complex.root.finding.png|240px|right|thumb|A quadratic function without real root: {{nowrap|''y'' {{=}} (''x'' − 5)<sup>2</sup> + 9}}. The "3" is the imaginary part of the ''x''-intercept. The real part is the ''x''-coordinate of the vertex. Thus the roots are {{nowrap|5 ± 3''i''}}.]]

The solutions of the quadratic equation 
<math display="block">ax^2+bx+c=0</math>
may be deduced from the [[graph of a function|graph]] of the [[quadratic function]]
<math display="block">f(x)=ax^2+bx+c,</math>
which is a [[parabola]].

If the parabola intersects the {{mvar|x}}-axis in two points, there are two real [[zero of a function|roots]], which are the {{mvar|x}}-coordinates of these two points (also called {{mvar|x}}-intercept).

If the parabola is [[tangent]] to the {{mvar|x}}-axis, there is a double root, which is the {{mvar|x}}-coordinate of the contact point between the graph and parabola.

If the parabola does not intersect the {{mvar|x}}-axis, there are two [[complex conjugate]] roots. Although these roots cannot be visualized on the graph, their [[complex number|real and imaginary parts]] can be.<ref name = "Norton1984">{{citation |title=Complex Roots Made Visible |author=Alec Norton, Benjamin Lotto |journal=The College Mathematics Journal |volume=15 |date=June 1984 |pages=248–249 |issue=3 |doi=10.2307/2686333|jstor=2686333 }}</ref>

Let {{mvar|h}} and {{mvar|k}} be respectively the {{mvar|x}}-coordinate and the {{mvar|y}}-coordinate of the vertex of the parabola (that is the point with maximal or minimal {{mvar|y}}-coordinate. The quadratic function may be rewritten
<math display="block"> y = a(x - h)^2 + k.</math>
Let {{mvar|d}} be the distance between the point of {{mvar|y}}-coordinate {{math|2''k''}} on the axis of the parabola, and a point on the parabola with the same {{mvar|y}}-coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is {{mvar|h}}, and their imaginary part are {{math|±''d''}}. That is, the roots are 
<math display="block">h+id \quad  \text{and} \quad h-id,</math>
or in the case of the example of the figure
<math display="block">5+3i \quad  \text{and} \quad 5-3i.</math>