To solve the given equation [tex]\(2x^2 + 3x + 7 = -3\)[/tex] and calculate the discriminant, we will follow these steps:
1. Move all terms to one side to set the equation to [tex]\(0\)[/tex].
The given equation is:
[tex]\[
2x^2 + 3x + 7 = -3
\][/tex]
Add 3 to both sides to get:
[tex]\[
2x^2 + 3x + 7 + 3 = 0
\][/tex]
Simplify the equation:
[tex]\[
2x^2 + 3x + 10 = 0
\][/tex]
2. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
The equation is now in standard form [tex]\(ax^2 + bx + c = 0\)[/tex]. From the equation [tex]\(2x^2 + 3x + 10 = 0\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 3\)[/tex]
- [tex]\(c = 10\)[/tex]
3. Calculate the discriminant.
The discriminant [tex]\(D\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[
D = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
D = 3^2 - 4(2)(10)
\][/tex]
Calculate [tex]\(b^2\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
Calculate [tex]\(4ac\)[/tex]:
[tex]\[
4 \cdot 2 \cdot 10 = 80
\][/tex]
Subtract [tex]\(4ac\)[/tex] from [tex]\(b^2\)[/tex]:
[tex]\[
D = 9 - 80 = -71
\][/tex]
Thus, the discriminant [tex]\(D\)[/tex] is:
[tex]\[
D = -71
\][/tex]
