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Copyright National Council of Teachers of Mathematics Dec 1997

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**Full Text**- Scholarly Journal

Goolsby, Ronnie C

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;Polaski, Thomas WPreview author details

. The Mathematics TeacherPreview publication details

**; Reston**Vol.90,Iss.9, (Dec 1997): 718-720.

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The widespread use of graphing calculators in both secondary and postsecondary mathematics allows students to investigate the standard topics of classic algebra in new and meaningful ways. The reasons behind certain algebraic techniques are sometimes difficult for average students to fathom; technology can help them visualize a result and thus understand and use the algebraic technique. Graphing can also be a source of qualitative information about the nature of the solutions to algebraic equations, and this information can help students check their solutions. As an example of using graphing to understand algebra, we investigate the solution of equations of the form ax+b = the square root of cx + d, where a, b, c, and d are constants. For convenience, we assume that a -= 0 and c -= 0. The figures in this article were generated by an HP 48G graphing calculator, but any currently available graphing calculator would be appropriate.

Let us first consider the solution of the equation the square root of 2x - 1 = 4-x.

The classic way to solve this equation is to square both sides of the equation, obtain a quadratic equation, and then either factor or use the quadratic formula to find candidates for a solution. The candidates are then checked to see whether they are solutions to the equations or are extraneous. However, graphing each side of this equation together with an auxiliary line can help a student realize before performing any algebra on the equation that this particular equation has one real solution, which must be between 1/2 and 4, and an extraneous solution, which is greater than 4.

First we graph the line y = 4 - x and the "half parabola" y = the square root of 2x -1, obtaining figure 1a. Since a solution to the equation...