To find the value of the discriminant for the quadratic equation [tex]\(0 = 2x^2 + x - 3\)[/tex], we need to use the discriminant formula [tex]\(D = b^2 - 4ac\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the equation [tex]\(ax^2 + bx + c = 0\)[/tex].
Given the quadratic equation:
[tex]\[ 2x^2 + x - 3 = 0 \][/tex]
We can identify the coefficients as follows:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 1\)[/tex]
- [tex]\(c = -3\)[/tex]
Now, let's calculate the discriminant step-by-step:
1. Square the coefficient [tex]\(b\)[/tex]:
[tex]\[ b^2 = 1^2 = 1 \][/tex]
2. Multiply 4 by [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
[tex]\[ 4ac = 4 \cdot 2 \cdot (-3) = 8 \cdot (-3) = -24 \][/tex]
3. Subtract [tex]\(4ac\)[/tex] from [tex]\(b^2\)[/tex]:
[tex]\[ b^2 - 4ac = 1 - (-24) = 1 + 24 = 25 \][/tex]
Therefore, the discriminant [tex]\(D\)[/tex] for the quadratic equation [tex]\(2x^2 + x - 3 = 0\)[/tex] is:
[tex]\[ D = 25 \][/tex]
Among the given options:
- [tex]\(-25\)[/tex]
- [tex]\(-23\)[/tex]
- [tex]\(25\)[/tex]
- [tex]\(26\)[/tex]
The correct answer is:
[tex]\[ 25 \][/tex]
