UNsolving Quadratic Equations & Inequalities

The solution of a quadratic equation is a pair of numbers - let's call them a and b [assume a smaller than or equal to b]. The solution of a quadratic inequality is EITHER all the numbers between a and b, OR all the numbers less than a AND all the numbers greater than b. To UNsolve a quadratic equation or inequality, drag the GOLD dots in the left panel to fix the solution set. The right hand panel will show you a quadratic equation or inequality that has that solution set. You can drag the WHITE dots in the right hand panel to see other quadratic equations or inequalities that have the same solution set.Each of the large WHITE dots control one function - the smaller WHITE dots control both functions. Why is it permissible to change only one function in an equation or inequality that is a comparison of two functions? For a given solution set, how many equations or inequalities are there that have that solution set? How do you know? Can you prove it?