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.
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