To determine the number of real zeros of the quadratic function
[tex]\[ y = 4x^2 - 12x + 9 \][/tex] we will use the discriminant method. The discriminant is found from the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] and is given by [tex]\( \Delta = b^2 - 4ac \)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are coefficients of the equation.
1. Identify the coefficients:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = -12\)[/tex]
- [tex]\(c = 9\)[/tex]
2. Substitute these values into the discriminant formula:
[tex]\[
\Delta = (-12)^2 - 4 \cdot 4 \cdot 9
\][/tex]
3. Calculate the squares and products step-by-step:
- [tex]\((-12)^2 = 144\)[/tex]
- [tex]\(4 \cdot 4 = 16\)[/tex]
- [tex]\(16 \cdot 9 = 144\)[/tex]
4. Substitute these results into the discriminant expression:
[tex]\[
\Delta = 144 - 144 = 0
\][/tex]
The discriminant [tex]\(\Delta\)[/tex] is 0.
- If [tex]\(\Delta > 0\)[/tex], the quadratic function has two distinct real zeros.
- If [tex]\(\Delta = 0\)[/tex], the quadratic function has exactly one real zero.
- If [tex]\(\Delta < 0\)[/tex], the quadratic function has no real zeros.
Since [tex]\(\Delta = 0\)[/tex]:
The quadratic function [tex]\( y = 4x^2 - 12x + 9 \)[/tex] has exactly one real zero.
