To determine the discriminant for the quadratic equation [tex]\(0 = 2x^2 + 3\)[/tex] and the number of real number solutions:
1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the standard form of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
Here, [tex]\(a = 2\)[/tex], [tex]\(b = 0\)[/tex], and [tex]\(c = 3\)[/tex].
2. The formula for the discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
3. Substitute the identified coefficients into the discriminant formula:
[tex]\[
\Delta = 0^2 - 4 \cdot 2 \cdot 3
\][/tex]
4. Simplify the expression:
[tex]\[
\Delta = 0 - 24 = -24
\][/tex]
Thus, the discriminant is [tex]\(-24\)[/tex].
5. Based on the value of the discriminant, we can determine the number of real number solutions for the quadratic equation:
- If the discriminant is greater than 0 ([tex]\(\Delta > 0\)[/tex]), there are two distinct real solutions.
- If the discriminant is equal to 0 ([tex]\(\Delta = 0\)[/tex]), there is exactly one real solution.
- If the discriminant is less than 0 ([tex]\(\Delta < 0\)[/tex]), there are no real solutions.
Since the discriminant [tex]\(\Delta = -24\)[/tex] is less than 0, the quadratic equation [tex]\(0 = 2x^2 + 3\)[/tex] has \emph{no real number solutions}.
Hence, the discriminant is [tex]\(-24\)[/tex], and the number of real number solutions is [tex]\(0\)[/tex].
