### Home > A2C > Chapter 9 > Lesson 9.2.2 > Problem 9-90

Raul claims that he has a shortcut for deciding what kind of roots a function has. Jolene thinks that a shortcut is not possible. She says you just have to solve the quadratic equation to find out. They are working on .

Jolene says, “*See, I just start out by trying to factor. This one can be factored* *, so the equation will have two real solutions and the function will have two real roots*.”

“*But what if it can't be factored*?” Raul asked. “*What about *?”

“*That's easy! I just use the Quadratic Formula,*” says Jolene. “*And I get… let's see… negative two plus or minus the square root of… two squared… that's 4… minus… eight…*”

“*Wait!*” Raul interrupted. “*Right there, see, you don't have to finish*. *minus* *, that gives you* *. that's all you need to know. You'll be taking the square root of a negative number so you will get a complex result*.”

“*Oh, I see,*” said Jolene. “*I only have to do part of the solution, the part you have to take the square root of*.”

Use Raul's method to tell whether each of the following functions has real or complex roots without completely solving the equation. Note: Raul's method is summarized in the Math Notes box for this lesson.

For the following, refer to the Math Note above from Lesson 9.2.2 (pg 460 in the student textbook).

; complex roots

Are the roots real or complex?