To solve this question, we need to find the discriminant of the quadratic equation and determine the number of real roots based on it.
The given quadratic equation is:
[tex]\[ x^2 + 3x + 8 = 0 \][/tex]
A quadratic equation is of the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
From the given equation, we can identify:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 3 \][/tex]
[tex]\[ c = 8 \][/tex]
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula, we get:
[tex]\[ \Delta = 3^2 - 4 \cdot 1 \cdot 8 \][/tex]
[tex]\[ \Delta = 9 - 32 \][/tex]
[tex]\[ \Delta = -23 \][/tex]
The value of the discriminant [tex]\(\Delta\)[/tex] is [tex]\(-23\)[/tex].
Next, we determine the number of real roots based on the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly one real root.
- If [tex]\(\Delta < 0\)[/tex], the equation has no real roots (the roots are complex).
Since [tex]\(\Delta = -23 < 0\)[/tex], the quadratic equation has no real roots.
Thus, the discriminant is [tex]\(-23\)[/tex] and the number of real roots is zero.
The correct answer is:
[tex]\[ \boxed{-23 ; no real roots} \][/tex]
