To analyze the quadratic function [tex]\( f(x) = -4x^2 + 10x - 8 \)[/tex], we need to follow these steps:
1. Identify the coefficients of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
a = -4, \quad b = 10, \quad c = -8
\][/tex]
2. Calculate the discriminant of the quadratic equation. The discriminant [tex]\(\Delta\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 10^2 - 4(-4)(-8)
\][/tex]
Simplify inside the parentheses and perform the calculations:
[tex]\[
\Delta = 100 - 4 \times (-4) \times (-8)
\][/tex]
[tex]\[
\Delta = 100 - 4 \times 16
\][/tex]
[tex]\[
\Delta = 100 - 64
\][/tex]
[tex]\[
\Delta = 36
\][/tex]
3. Determine the number of distinct real number zeros based on the discriminant:
- 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 numbers).
From the calculation above, we found that:
[tex]\[
\Delta = -28
\][/tex]
Since [tex]\(\Delta < 0\)[/tex], this indicates that there are no distinct real number zeros for the given quadratic function.
Therefore:
- The value of the discriminant of [tex]\( f \)[/tex] is [tex]\(-28\)[/tex].
- The number of distinct real number zeros [tex]\( f \)[/tex] has is [tex]\( 0 \)[/tex].
