Question

f(x) = 3x^2+12x+5
What is the value of the discriminant of F
How many distinct real number zeros does \[f(x)\] have?

Answer 1

To find the value of the discriminant of the quadratic function \( f(x) = 3x^2 + 12x + 5 \), we can use the formula for the discriminant, which is given by:

\[
D = b^2 - 4ac
\]

In this case, \( a = 3 \), \( b = 12 \), and \( c = 5 \).

Now, we can plug these values into the formula:

\[
D = (12)^2 - 4 \cdot 3 \cdot 5
\]
\[
D = 144 - 60
\]
\[
D = 84
\]

The discriminant \( D = 84 \).

Now, we can determine the number of distinct real number zeros based on the value of the discriminant:
- If \( D > 0 \), there are two distinct real zeros.
- If \( D = 0 \), there is one real zero (a repeated root).
- If \( D < 0 \), there are no real zeros.

Since \( D = 84 > 0 \), we conclude that the function \( f(x) \) has two distinct real number zeros.

**Summary:**
- The value of the discriminant is \( 84 \).
- There are two distinct real number zeros.