# Activities

• • • ##### Subject Area

• Math: Algebra I: Equations and Inequalities

• ##### Author 9-12

45 Minutes

Derive™ 6

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Finding the Roots of a Cubic Equation - An Excursion into Mathematics History

#### Activity Overview

In this Derive™ activity, students understand how the software solves cubic equations and relate it to contributions made by del Ferro, Tartaglia, and Cardan in finding the method of solving cubic equations.

#### Before the Activity

• See the attached DFW file for detailed instructions for this activity
• Print pages from the attached DFW file for your class
• #### During the Activity

Distribute the pages to the class.

• Set up a 2D plot window
• Enter the standard cubic equation and choose random values for its coefficients
• Continue simplifying until values obtained for the variable, coeff, contains all non zero values

• Define a function and expand it so that the function is seen as a cubic
• Adjust the scale and plot a graph that includes both the maximum and minimum values of the function

• Shift the axes, and eliminate the squared term in the cubic equation by making a substitution for x in terms of another variable
• Expand the result in terms of the new variable
• Find the value which causes the coefficient of the squared term to be zero
• Substitute the value in the expression and observe that the resulting cubic in the new variable has no squared term
• Substitute the new variable with yet another quantity
• Simplify the expression, work backwards, and determine the value of x
• By factoring, find all the values of x that satisfy the standard cubic expression
• #### After the Activity

Students answer questions listed on the activity sheet.

Review student results:

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary