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Sagot :
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.
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.
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