A quadratic equation is always of the form [math]f(x)=g(x)[/math]. For example, in the equation [br][math]ax^2+bx+c=Ax^2+Bx+C[/math] we can regard [math] f(x)[/math] as [math]ax^2+bx+c[/math] and [math]g(x)[/math] as [math]Ax^2+Bx+C[/math].[br][br]Solving a quadratic equation means transforming the original equation into a new equation that has the form [math](x-P)^2 = Q^2[/math]. We can then take the square root of both sides of the equation and get [math]x = P+Q[/math] and [math]x = P-Q[/math][br][br]The graph of the function [math](x-P)^2[/math] is a parabola that is open (concave) upward and just touches (tangent to) the x-axis at [math]P[/math]. The graph of the constant function [math]Q^2[/math] is a horizontal line above and parallel to the x-axis.[br][br]This applet allows you to enter a quadratic function [math]f(x)[/math] by varying a, b and c sliders and a function [math]g(x)[/math] by varying A, B and C sliders.[br][br]You may solve your equation [size=150][u][i][b]graphically[/b][/i][/u][/size] by dragging the [color=#00ff00][i][b]GREEN[/b][/i][/color], [color=#1e84cc][i][b]BLUE[/b][/i][/color] and [i][b]WHITE[/b][/i] dots on the graph in order to produce a 'solution equation' of the form [math](x-P)^2 = Q^2.[/math][math][/math] The solution set of the equation can then be gotten by taking the square root of both sides.[br][br][i][b][size=150]Challenge[/size][/b][/i] - Dragging the [i][b]WHITE[/b][/i] dots changes both functions, but dragging the [color=#00ff00][i][b]GREEN[/b][/i][/color] dot changes only the [color=#00ff00][i][b]GREEN[/b][/i][/color] function and dragging the [color=#1e84cc][i][b]BLUE[/b][/i][/color] dot changes only the [color=#1e84cc][i][b]BLUE[/b][/i][/color] function.[br][br]This means that when you drag either the [color=#00ff00][i][b]GREEN[/b][/i][/color] dot or the [color=#1e84cc][i][b]BLUE[/b][/i][/color] dot you are changing only [b][i]ONE[/i][/b] side of the equation!! [br] [br]- Why is this legitimate? [br] [br]- Why are we taught that you must do the same thing to both sides of the equation? [br] [br]- What is true about all the legitimate things you can do to a quadratic equation?[br][br]You can also solve your quadratic equation [size=150][u][i][b]symbolically[/b][/i][/u][/size] by using sliders. The sliders allow you to change the constant term, the linear term and the quadratic term on both sides of the equation as well as to scale both sides of the equation.[br][br]- What symbolic operations correspond to the actions you take when solving graphically? [br][br]- What graphical operations correspond to the actions you take when solving graphically?[br][br][i][b][color=#ff0000]What other questions could/would you pose to your students based on this applet ?[/color][/b][/i]