1. For the trinomial x^2+5x+1, find at least six different expressions that can be added to the trinomial to make it a perfect square trinomial. You will need to think outside the box of the usual “complete the square” algorithm. Use the Algebra Tiles applet at http://nlvm.usu.edu/en/nav/frames_asid_189_g_4_t_2.html?open=activities&from=category_g_4_t_2.html to explore different ways to make a square starting with one “x2” tile, five “x” tiles, and one “1” tile.[br][br]2. Identify a general formula for the completer expression you need to add to the seed trinomial x^2+5x+1 so that it can be factored as (x+n)^2 .[br][br]3. Identify a general formula for the completer expression you need to add to the seed trinomial ax^2+bx+c so that it can be factored as (x+n)^2 .[br][br]Instructions for Constructing GeoGebra File below:[br]1. Open a new GeoGebra file. Create three sliders named a, b, and c. Set each one to vary between -10 and 10 at increments of 1. To begin, set a=1, b=5, and c=1.[br]2. Make the Spreadsheet view visible. Enter the number -20 in cell A1. In cell A2 type =A1+1. Drag and copy the contents of this cell into cells A3 through A40.[br]3. In column A, we are going to calculate the slope of each completer expression. In cell B1 type =2*A1-b. Copy and paste this into cells B2 through B40.[br]4. In column C, we are going to calculate the intercept of each completer expression. In cell C1 type =A1^2-c. Drag to copy this into cells C2 through C40.[br]5. In column D, we are going to enter the completer expression. In cell D1 type =B1*x+C1. Drag to copy this into cells D2 through D40.[br]6. Add the seed trinomial to the graph. Type f1(x)=a*x^2+b*x+c into the Input bar. Color it and make it thick so it stands out.[br]7. Change the sliders for b and c. What do you notice?[br]8. Construct the visible intersection points for consecutive completer functions.[br]9. Create a list of these points. For example: L={A,B,C,D,E,F,G,H,I}.[br]10. Use the FitPoly function to fit a quadratic function to these points. FitPoly[L,2].[br]11. Develop an expression for this new quadratic function in terms of the seed trinomial.

Extension questions to take this idea further:[br]1. Graph the negatives of each completer expression. Show that each one is tangent to the seed trinomial at x = n.[br]2. Start with the seed trinomial . Derive a general formula for the completer expression that will allow the trinomial to be factored as .[br]3. Now derive a general formula for the completer expression that will allow any trinomial of the form to be factored as .[br]4. Reconstruct the GeoGebra file to graph these new completer expressions. [br][br]References:[br]Phelps, S. a. (2010). New Life for an Old Topic: Completing the Square Using Technology. [i]Mathematics Teacher[/i], 230-236.