To determine the number of real solutions for the quadratic equation [tex]\(0 = 2x^2 + 3\)[/tex], we first need to calculate its discriminant. The general form for a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. In our case, the coefficients are:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 0\)[/tex]
- [tex]\(c = 3\)[/tex]
The discriminant [tex]\(\Delta\)[/tex] of the quadratic equation 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]\[
\Delta = (0)^2 - 4 \cdot 2 \cdot 3
\][/tex]
Simplify the expression:
[tex]\[
\Delta = 0 - 4 \cdot 2 \cdot 3 = 0 - 24 = -24
\][/tex]
The discriminant [tex]\(\Delta\)[/tex] is [tex]\(-24\)[/tex].
The next step is to determine the number of real solutions based on the value of the discriminant:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], there is exactly one real solution.
- If [tex]\(\Delta < 0\)[/tex], there are no real solutions.
Since [tex]\(\Delta = -24\)[/tex], which is less than 0, the quadratic equation [tex]\(0 = 2x^2 + 3\)[/tex] has no real number solutions.
Therefore, the discriminant is [tex]\(-24\)[/tex], and the equation has [tex]\(0\)[/tex] real number solutions.
