The purpose of this discussion is for students to see the connection between the factored and standard forms and recognize that the family of functions that have the same two zeros have equations that differ only by a constant multiple.

Select groups to share their solutions. Display the equation for all to see. 

Highlight the solution to the last question and note that all quadratic functions with zeros at 3 and 4 are of the form . 

Display the functions and . Ask students:

• “Do the two functions have the same zeros? Explain your reasoning.” (Yes. Each function is 0 when and when .)
• “What is another quadratic function that has zeros when and when ?” (any constant multiple of , such as )
• “Are the functions equivalent?” (No, because when , for example, the first function is and the second function is .)

In this activity, students reconstruct the standard form of a quadratic equation from the solutions guaranteed by the quadratic formula. The activity gives students an opportunity to practice rewriting expressions that include a square root and see how some complex-looking expressions can be rewritten into simpler forms.

Students must use the structure of the zeros to find ways to rewrite the expressions in simpler forms (MP7).

In this activity, students use the difference of squares identity to rewrite expressions that have a square root.

Because calculators can be found readily, rationalizing a denominator is not as important of a skill as it once was. On the other hand, the technique of multiplying an expression by a version of 1 to change the form of the expression without changing its value is useful in later mathematics courses. The identities (when ) and may seem trivial to students but are useful for rewriting expressions.