To find the discriminant of the quadratic function \( f(x) = 4x^2 + 2x + 6 \), we can use the formula for the discriminant of a quadratic equation \( ax^2 + bx + c \), which is given by:
\[ D = b^2 - 4ac \]
In this case, the coefficients are:
- \( a = 4 \)
- \( b = 2 \)
- \( c = 6 \)
Now we can calculate the discriminant:
\[ D = (2)^2 - 4 \cdot (4) \cdot (6) \] \[ D = 4 - 96 \] \[ D = -92 \]
The discriminant \( D \) is \(-92\).
Now, to determine how many distinct real number zeros the function \( f \) has, we examine the value of the discriminant:
- If \( D > 0 \): There are 2 distinct real roots.
- If \( D = 0 \): There is exactly 1 real root (a double root).
- If \( D < 0 \): There are no real roots.
Since \( D = -92 < 0 \), we conclude that the function \( f(x) = 4x^2 + 2x + 6 \) has no distinct real number zeros.
Summary:
- The value of the discriminant is \(-92\).
- The function \( f \) has 0 distinct real number zeros.
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