To find the real solutions of the quadratic equation [tex]\(3x^2 + 3x + 8 = 0\)[/tex], we can follow these steps:
1. Identify the coefficients:
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, we have:
[tex]\[
a = 3, \quad b = 3, \quad c = 8
\][/tex]
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 3^2 - 4 \cdot 3 \cdot 8
\][/tex]
Simplify it step by step:
[tex]\[
\Delta = 9 - 96 = -87
\][/tex]
3. Analyze the discriminant:
The nature of the roots of the quadratic equation depends on the discriminant:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], there is one real solution (a repeated root).
- If [tex]\(\Delta < 0\)[/tex], there are no real solutions; the solutions are complex.
Here, the discriminant [tex]\(\Delta = -87\)[/tex] is less than zero. This indicates that there are no real solutions.
Therefore, the quadratic equation [tex]\(3x^2 + 3x + 8 = 0\)[/tex] does not have any real solutions.
