To determine the number of zeros (roots) of the quadratic equation [tex]\(0 = 3x^2 - 7x + 4\)[/tex], we need to use the discriminant of the quadratic formula.
The general form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. For the given equation:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = 4\)[/tex]
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is calculated using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Let's substitute the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into this formula:
1. Calculate [tex]\(b^2\)[/tex]:
[tex]\[ (-7)^2 = 49 \][/tex]
2. Calculate [tex]\(4ac\)[/tex]:
[tex]\[ 4 \cdot 3 \cdot 4 = 48 \][/tex]
3. Compute the discriminant:
[tex]\[ \Delta = 49 - 48 = 1 \][/tex]
Now that we have the discriminant [tex]\(\Delta = 1\)[/tex], we can determine the number of zeros:
- If the discriminant ([tex]\(\Delta\)[/tex]) is greater than 0, the quadratic equation has two real and distinct zeros.
- If the discriminant ([tex]\(\Delta\)[/tex]) is equal to 0, the quadratic equation has exactly one real zero (a repeated root).
- If the discriminant ([tex]\(\Delta\)[/tex]) is less than 0, the quadratic equation has no real zeros (the roots are complex).
Since our discriminant [tex]\(\Delta = 1\)[/tex] is greater than 0, this means:
- The equation has two real and distinct zeros.
Therefore, the best description of the discriminant and the number of zeros is:
[tex]\[ \text{The equation has two zeros because the discriminant is greater than 0.} \][/tex]
