To determine which quadratic equation has a discriminant of 12, we will calculate and analyze the discriminants of each given equation. 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]
### Equation 1: [tex]\(0 = -x^2 + 8x + 2\)[/tex]
In this equation, [tex]\(a = -1\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = 2\)[/tex]. The discriminant is calculated as follows:
[tex]\[
\Delta = b^2 - 4ac = 8^2 - 4(-1)(2) = 64 + 8 = 72
\][/tex]
So, the discriminant of the first equation is 72.
### Equation 2: [tex]\(0 = 2x^2 + 6x + 3\)[/tex]
In this equation, [tex]\(a = 2\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = 3\)[/tex]. The discriminant is calculated as follows:
[tex]\[
\Delta = b^2 - 4ac = 6^2 - 4(2)(3) = 36 - 24 = 12
\][/tex]
So, the discriminant of the second equation is 12.
### Equation 3: [tex]\(0 = -x^2 + 4x + 1\)[/tex]
In this equation, [tex]\(a = -1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = 1\)[/tex]. The discriminant is calculated as follows:
[tex]\[
\Delta = b^2 - 4ac = 4^2 - 4(-1)(1) = 16 + 4 = 20
\][/tex]
So, the discriminant of the third equation is 20.
### Equation 4: [tex]\(0 = 4x^2 + 2x + 1\)[/tex]
In this equation, [tex]\(a = 4\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = 1\)[/tex]. The discriminant is calculated as follows:
[tex]\[
\Delta = b^2 - 4ac = 2^2 - 4(4)(1) = 4 - 16 = -12
\][/tex]
So, the discriminant of the fourth equation is -12.
### Conclusion
By comparing the discriminants calculated for each equation with the given discriminant of 12, we find that the quadratic equation
[tex]\[
0 = 2x^2 + 6x + 3
\][/tex]
has a discriminant of 12. Hence, this is the equation that could have a discriminant of 12.
