# Completing the Square and Then Some

- Author:
- Camille Fairbourn

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.
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 .
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 .
Instructions for Constructing GeoGebra File below:
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.
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.
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.
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.
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.
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.
7. Change the sliders for b and c. What do you notice?
8. Construct the visible intersection points for consecutive completer functions.
9. Create a list of these points. For example: L={A,B,C,D,E,F,G,H,I}.
10. Use the FitPoly function to fit a quadratic function to these points. FitPoly[L,2].
11. Develop an expression for this new quadratic function in terms of the seed trinomial.

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

*Mathematics Teacher*, 230-236.