A quadratic equation is always of the form f(x) = g(x). For example, in the equation
we can regard f(x) as and g(x) as
Solving a quadratic equation means transforming the original equation into a new equation that has the
form (where Q is a constant). We can then take the square root of both sides of the equation and get
The graph of the function is a parabola that is open (concave) upward and just touches (tangent to) the x-axis at x = P. The graph of the constant function Q^2 is a horizontal line above and parallel to the x-axis.
The environment allows you to enter a quadratic function f(x) = by varying a, b and c sliders and a function g(x) = by varying A, B and C sliders.
You may solve your equation by dragging the RED, BLUE and BLACK dots on the graph in order to produce a 'solution equation' of the form . The solution set of the equation can then be gotten by taking the square root of both sides yielding
Challenge - Dragging the BLACK dots changes both functions, but dragging the RED dot changes only the RED function and dragging the BLUE dot changes only the BLUE function.
This means that when you drag either the RED dot or the BLUE dot you are changing only one side of the equation!! Why is this legitimate? Why are we taught that you must do the same thing to both sides of the equation? What is true about all the legitimate things you can do to a quadratic equation?
What do the solution sets of quadratic inequalities look like?