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 $y = x^2 - 5x - 14$.

Jolene says, “See, I just start out by trying to factor. This one can be factored $(x - 7)(x + 2) = 0$, 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 $x^2 + 2x + 2 = 0$?”

“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. $2^2$ minus $4 · 2$, that gives you $-4$. 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 consider part of the solution, the inside of the square root.”

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 also summarized in the Math Notes box for this lesson.