Question

How many x-intercepts does $f\left(x\right)=x^2+3x+3$ have?

• A. $1$
• B. $0$
• C. $2$
• D. $3$

Answer 1

Answer: (B)

$0$

The method is to use the discriminant of the quadratic equation to count the number of x-intercepts.

First, identify the coefficients of each term. The function is $f\left(x\right)=x^2+3x+3$. Matching this up to the definition of the standard quadratic equation, $f\left(x\right)=a{x}^2+bx+c$, you have $a=1,b=3$, and $c=3$.

Next, write the discriminant from the quadratic formula (the expression that is under the radical sign in the quadratic formula):

$b^2+4ac=3^2-4\left(1\right)\left(3\right)$

$=9-12$

$=-3$

Since the discriminant is less than 0, you cannot take its square root. This means that there is no solution to the equation $f\left(x\right)=x^2+3x+3=0$, so the function's graph does not touch the x-axis, There are no x-intercepts,