Video: Finding the Set of Zeros of a Quadratic Function

Video Transcript

What are the zeros of 𝑓 of 𝑥 equal to 𝑥 plus 𝜋 squared minus 𝑒?

First, we copy down our problem. And then in place of 𝑓 of 𝑥, we want to put a zero. We’re looking for the place where 𝑓 of 𝑥 is equal to zero. And then, we can solve this using algebra. We’re trying to isolate 𝑥, get 𝑥 by itself.

The first thing I can do is get rid of the 𝑒 by adding 𝑒 to the right side of the equation. If we add 𝑒 to the right, we have to add 𝑒 to the left. Negative 𝑒 plus 𝑒 equals zero. Zero plus 𝑒 equals 𝑒. Our equation now says 𝑒 is equal to 𝑥 plus 𝜋 squared. To get rid of the square, we take the square root of 𝑥 plus 𝜋 squared.

But if we take the square root on the right, we need to take the square root on the left. On the right side of the equation, the only thing that remains is 𝑥 plus 𝜋. The left side is a little bit different. On the left side, we need to say that we have two options: we have the positive square root of 𝑒 and the negative square root of 𝑒. We have two cases.

There is no reason for our parentheses anymore. So we can drop those and then we subtract 𝜋 from both sides of the equation. Positive 𝜋 minus 𝜋 equals zero. 𝑥 is by itself on the right side. On the left side, there is nothing really that we can simplify. We can just say negative 𝜋 plus or minus the square root of 𝑒.

And this is how we break that down into two different options: 𝑥 could be equal to negative 𝜋 plus 𝑒 or negative 𝜋 minus the square root of 𝑒.

This function has two zeros: negative 𝜋 plus the square root of 𝑒 and negative 𝜋 minus the square root of 𝑒.