Question

f(x) = 3x^2+24x+48
What is the value of the discriminant of F
How many distinct real number zeros does
\[f\] have?

Answer 1

To determine the value of the discriminant of the quadratic function \( f(x) = 3x^2 + 24x + 48 \), we first need to identify the coefficients of the quadratic equation in the standard form \( ax^2 + bx + c \):

- \( a = 3 \)
- \( b = 24 \)
- \( c = 48 \)

The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c \) is given by the formula:

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

Now, we can substitute the values of \( a \), \( b \), and \( c \) into the formula:

\[
D = (24)^2 - 4(3)(48)
\]

Calculating \( (24)^2 \):

\[
24^2 = 576
\]

Now we calculate \( 4ac \):

\[
4ac = 4 \times 3 \times 48 = 576
\]

Now, we substitute these values back into the discriminant formula:

\[
D = 576 - 576 = 0
\]

The discriminant \( D \) is \( 0 \).

The number of distinct real number zeros of the quadratic function can be determined by the value of the discriminant:

- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is exactly one distinct real root (a repeated root).
- If \( D < 0 \), there are no real roots.

Since we found that \( D = 0 \), it indicates that the quadratic function \( f(x) \) has exactly one distinct real root.

**Summary:**
- The value of the discriminant is \( 0 \).
- The function \( f(x) \) has exactly one distinct real number zero.