To determine the number of real number solutions for the quadratic equation [tex]\(0 = -2x^2 + 3\)[/tex], we need to calculate the discriminant and then use the value of the discriminant to determine the number of solutions. Here are the steps:
1. Identify the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
For the given equation [tex]\(0 = -2x^2 + 3\)[/tex]:
[tex]\[
a = -2, \quad b = 0, \quad c = 3
\][/tex]
2. Write down the formula for the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
3. Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\Delta = 0^2 - 4(-2)(3)
\][/tex]
4. Simplify the expression:
[tex]\[
\Delta = 0 - (-24)
\][/tex]
[tex]\[
\Delta = 24
\][/tex]
5. Interpret the value of the discriminant:
The value of the discriminant [tex]\(\Delta\)[/tex] determines the number of real solutions as follows:
- If [tex]\(\Delta > 0\)[/tex], there are 2 distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], there is exactly 1 real solution (a repeated root).
- If [tex]\(\Delta < 0\)[/tex], there are no real solutions (the solutions are complex).
For our equation, [tex]\(\Delta = 24 > 0\)[/tex].
6. Conclusion:
Since the discriminant [tex]\(\Delta = 24\)[/tex] is greater than zero, the quadratic equation [tex]\(0 = -2x^2 + 3\)[/tex] has 2 distinct real solutions.
Therefore, the discriminant for the quadratic equation is 24, and the equation has 2 real number solutions.
