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Instructions: Put the equation in standard form, then calculate the discriminant.

[tex]\[
\begin{array}{l}
2x^2 + 3x + 7 = -3 \\
\text{Standard form: } 2x^2 + 3x + 10 = 0 \\
D = \square
\end{array}
\][/tex]

Note: To find the discriminant [tex]\(D\)[/tex], use the formula [tex]\(D = b^2 - 4ac\)[/tex] where the equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex].


Sagot :

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]
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