For what values of is the function equal to 0? Recall that these values are called zeros.
Rewrite the equation in standard form, .
For this function, how are and related to the zeros?
Is this the only quadratic function with zeros at 3 and 4?
Let .
What are the zeros of ?
Rewrite the equation in standard form.
13.2
Activity
An Identity from Zeros
Let and be the two solutions from the quadratic formula.
Multiply and to find a simple expression in terms of , , and .
Add and to find a simple expression in terms of , , and .
Show that .
Use the connection between the variables , , , and to write the quadratic function in standard form that has zeros at and and a quadratic coefficient of 1 (). Explain or show your reasoning.
13.3
Activity
Removing Square Roots
Finish this identity: . Are there any restrictions on what can be for the identity to be true?
Finish this identity: . Are there any restrictions on what can be for the identity to be true?
Pause here for a discussion.
Show that
Show that
Student Lesson Summary
Mathematical identities can be useful in a variety of situations.
In an earlier course, we learned how to derive the quadratic formula using the method of completing the square on a quadratic equation. We can check the formula by reconstructing the standard form of a quadratic equation using the zeros guaranteed by the quadratic formula. We can start with writing the factored form using the zeros as .
After distributing the left side of the equation, it nicely becomes .
Identities can also be used to rewrite expressions to be in a form that is clearer to understand. For example, the value of the fraction is not very easy to estimate without a calculator. Let’s rewrite the expression using two identities: the difference of squares identity, , and the identity when .
In this form, we can estimate that is approximately 2.5 and is a little more than 1, so the expression has a value of approximately 1.25. Checking on a calculator, we can see that the original fraction and the final expression are equal and have a value of approximately 1.232, which is close to our estimation.
