Let's solve the problem step-by-step:
1. Identify the coefficients of the quadratic equation:
[tex]\[
x^2 + 2x + 3 = 0
\][/tex]
Here, the coefficients are:
[tex]\[
a = 1, \quad b = 2, \quad c = 3
\][/tex]
2. Calculate the discriminant [tex]\(\Delta\)[/tex] using the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[
\Delta = 2^2 - 4 \cdot 1 \cdot 3
\][/tex]
3. Simplify the expression:
[tex]\[
\Delta = 4 - 12
\][/tex]
[tex]\[
\Delta = -8
\][/tex]
4. Interpret the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the quadratic equation has two different real-number solutions.
- If [tex]\(\Delta = 0\)[/tex], the quadratic equation has one real-number solution.
- If [tex]\(\Delta < 0\)[/tex], the quadratic equation has two different imaginary-number solutions.
Since the discriminant [tex]\(\Delta = -8\)[/tex], which is less than zero, the quadratic equation:
[tex]\[
x^2 + 2x + 3 = 0
\][/tex]
has two different imaginary-number solutions.
Therefore, the discriminant is [tex]\(-8\)[/tex], and the quadratic equation has two different imaginary-number solutions.
