Quadratic functions - Completing the Square

This applet investigates the graphs and techniques of "completing the square", to solve quadratic equations, sketch quadratic graphs and find the coordinates of the vertex. This technique is widely used in mathematics, including finding the Cartesian Equation of a Circle.

Author and programmer: Ron Barrow

UK Years 12-13, KS5, KS4, Core 1 (C1) GCE Mathematics, Algebra and Functionsd
US - Grades 11 - 12


Instructions below   See also:    Quadratics:The Discriminant   Quadratic Inverses   Quadratic Inequalities  
Equations of Circles

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How to Use this Applet

The applet on this page is designed to help you understand how to "complete the square" when dealing with quadratic equations and expressions. The standard quadratic expression is: ax² + bx + c. Completing the square enables you to rewrite this expression in the form: a(x + p)² + q. Varying a, b, c changes the curves. As the curve changes, notice how the working out of the completed-square form changes. It's a good to idea to keep a = 1 to begin with. The completed-square form is useful in several ways, including:
1. It gives the coordinates of the vertex of the parabola. This is shown at the bottom right of the screen.
2. It is another method of solving quadratic equations.
Study this applet carefully and you will soon be familiar with this very important mathematical technique, which has many other applications.