To find the discriminant and the number of distinct real number zeros of the quadratic function [tex]\( f(x) = 4x^2 - 17x + 3 \)[/tex]:
1. Identify the coefficients: For the quadratic equation in the standard form [tex]\( ax^2 + bx + c \)[/tex], the coefficients are:
[tex]\[
a = 4, \quad b = -17, \quad c = 3
\][/tex]
2. Calculate the discriminant: The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the coefficients [tex]\( a = 4 \)[/tex], [tex]\( b = -17 \)[/tex], and [tex]\( c = 3 \)[/tex] into the formula, the discriminant is:
[tex]\[
\Delta = (-17)^2 - 4 \cdot 4 \cdot 3 = 289 - 48 = 241
\][/tex]
So, the value of the discriminant is:
[tex]\[
\boxed{241}
\][/tex]
3. Determine the number of distinct real number zeros: The number of distinct real zeros of the quadratic equation depends on the value of the discriminant [tex]\( \Delta \)[/tex]:
- If [tex]\( \Delta > 0 \)[/tex], there are 2 distinct real zeros.
- If [tex]\( \Delta = 0 \)[/tex], there is 1 distinct real zero.
- If [tex]\( \Delta < 0 \)[/tex], there are no real zeros (the zeros are complex).
Since the discriminant [tex]\( \Delta = 241 \)[/tex] is greater than 0, this quadratic function has:
[tex]\[
\boxed{2}
\][/tex]
distinct real number zeros.
