Editing Quadratic equation (section)

====Geometric solution====
[[File:LillsQuadratic.svg|thumb|180px|Figure 6. Geometric solution of {{math|''ax''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}} using Lill's method. Solutions are −AX1/SA, −AX2/SA|alt=Figure 6. Geometric solution of eh x squared plus b x plus c = 0 using Lill's method. The geometric construction is as follows: Draw a trapezoid S Eh B C. Line S Eh of length eh is the vertical left side of the trapezoid. Line Eh B of length b is the horizontal bottom of the trapezoid. Line B C of length c is the vertical right side of the trapezoid. Line C S completes the trapezoid. From the midpoint of line C S, draw a circle passing through points C and S. Depending on the relative lengths of eh, b, and c, the circle may or may not intersect line Eh B. If it does, then the equation has a solution. If we call the intersection points X 1 and X 2, then the two solutions are given by negative Eh X 1 divided by S Eh, and negative Eh X 2 divided by S Eh.]]

The quadratic equation may be solved geometrically in a number of ways. One way is via [[Lill's method]]. The three coefficients {{math|''a''}}, {{math|''b''}}, {{math|''c''}} are drawn with right angles between them as in SA, AB, and BC in Figure&nbsp;6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient {{math|''a''}} or SA. If {{math|''a''}} is {{math|1}} the coefficients may be read off directly. Thus the solutions in the diagram are &minus;AX1/SA and &minus;AX2/SA.<ref>{{Citation |title=Graphical Method for finding readily the Real Roots of Numerical Equations of Any Degree |first=William Herbert |last=Bixby |year=1879 |publisher=West Point N. Y.}}</ref>

[[File:CarlyleCircle.svg|thumb|300px|left|Carlyle circle of the quadratic equation ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''sx''&nbsp;+&nbsp;''p''&nbsp;=&nbsp;0.]]
The [[Carlyle circle]], named after [[Thomas Carlyle]], has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the [[horizontal axis]].<ref name=Wolfram>{{cite web|last=Weisstein|first=Eric W|title=Carlyle Circle|url=http://mathworld.wolfram.com/CarlyleCircle.html|work=From MathWorld—A Wolfram Web Resource|access-date=21 May 2013}}</ref> Carlyle circles have been used to develop [[ruler-and-compass construction]]s of [[regular polygon]]s.