## Quadratic Equations and Completing the Square

This applet explores quadratic equations, linking the algebraic methods with corresponding geometric interpretations.

The window is set up with sliders for a, b, and c,  in the quadratic equation y=a*x^2+b*x+c.  The graph of the function is also given.

One algebraic approach to the equation is to look for roots using the quadratic equation.  For that we compute the discriminant,
D=b^2-4*a*c, then us that to compute the roots, one red and one green.  Notice that if the discriminant is negative, the equation has no roots, if the discriminant is zero the equation has one root, and if the discriminant is positive, the equation has two roots.  If the equation can be solved, we also want to look at the equation in factored form.

A second algebraic approach uses the method of completing the square to find the vertex of the parabola.  This is given in blue.  If the vertex is the point (h,k), then the equation can also be written in the form y = a(x-h)^2 + k.

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The method of completing the square deserves more attention, so we look at it in more detail in the next window.  Once again, sliders let  us control a, b, and c, and  we can see how the method of completing the square works out for various quadratic polynomials

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Mike May, S.J., 5/7/06, Created with GeoGebra

GeoGebra is a GNUed software package for mathematics visualization.  The home for the applications is  http://www.geogebra.at.