To find the discriminant of the given quadratic equation [tex]\(0 = -2x^2 + 6x + 3\)[/tex], we use the formula for the discriminant of any quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], which is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Here, the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] can be identified from the given quadratic equation:
- [tex]\(a = -2\)[/tex]
- [tex]\(b = 6\)[/tex]
- [tex]\(c = 3\)[/tex]
Now substitute these values into the discriminant formula:
[tex]\[ \Delta = 6^2 - 4 \times (-2) \times 3 \][/tex]
First, calculate [tex]\(6^2\)[/tex]:
[tex]\[ 6^2 = 36 \][/tex]
Next, calculate [tex]\(-4 \times (-2) \times 3\)[/tex]:
[tex]\[ -4 \times -2 = 8 \][/tex]
[tex]\[ 8 \times 3 = 24 \][/tex]
So, the discriminant becomes:
[tex]\[ \Delta = 36 + 24 \][/tex]
Finally, add them together:
[tex]\[ 36 + 24 = 60 \][/tex]
Therefore, the value of the discriminant for the quadratic equation [tex]\(0 = -2x^2 + 6x + 3\)[/tex] is:
[tex]\[ \Delta = 60 \][/tex]
