When you solve the following equation, you get 2 solutions. At least that is what it looks like initially. But then when you substitute the values into the equation, only one works! Why does that happen? What is going on here?
In years past, I would have said something to the effect that squaring in this particular problem made a new equation with two solutions and yet one of the solutions was not a solution to the original equation. Although this was true, I would often get blank stares from my students.
Today someone asked me this question and in an instant I was able to show the following two screenshots from Desmos.
By graphing both sides of the original equation, we can see that there is only one solution, x = 6. Notice that squaring both of these expressions turns the red line into a parabola and turns the blue square root function into a line. We can see that the line and the parabola have two points of intersection, when x = 3 AND when x = 6. Transforming the equation by squaring both sides of the equation changed the graphs in such a way that an additional solution was created that was not a solution to the original equation.
Connecting ideas in mathematics was something that I don't recall from my days as a high school student or from the early days of my teaching career. Now, it is so quick and easy to help students to form those connections with tools, like Desmos.
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