Let's solve the quadratic function $f(x) = x^2 + 6x + 6$ using the quadratic formula to find the x-intercepts.

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $f(x) = x^2 + 6x + 6$, we have:

• $a = 1$
• $b = 6$
• $c = 6$

Now, let's substitute these values into the quadratic formula.

Step 1: Compute the discriminant

$b^2 - 4ac = 6^2 - 4(1)(6) = 36 - 24 = 12$

Step 2: Apply the quadratic formula

$x = \frac{-6 \pm \sqrt{12}}{2(1)} = \frac{-6 \pm 2\sqrt{3}}{2}$ Simplify: $x = -3 \pm \sqrt{3}$

Solution:

The zeros of the function are $x = -3 + \sqrt{3}$ and $x = -3 - \sqrt{3}$.

Thus, the zeros and x-intercepts are the same, and the correct choice is B.

Would you like a more detailed explanation of any part of this solution?

Here are 5 related questions you might find interesting:

1. How does the discriminant affect the number of real solutions in a quadratic equation?
2. What does it mean if the discriminant is negative in a quadratic equation?
3. How would the solution change if the quadratic had a higher degree, say cubic?
4. Can you solve a quadratic equation without using the quadratic formula?
5. How can completing the square be used to solve a quadratic function?

Tip: When solving quadratics, always check the discriminant first to quickly determine the nature of the roots!