To find the discriminant and number of real roots for the quadratic equation [tex]\(x^2 + 3x + 8 = 0\)[/tex], follow the steps below:
### Step 1: Identify the Coefficients
In the quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\(a\)[/tex] is the coefficient of [tex]\(x^2\)[/tex]
- [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]
- [tex]\(c\)[/tex] is the constant term
For the given equation [tex]\(x^2 + 3x + 8 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 3\)[/tex]
- [tex]\(c = 8\)[/tex]
### Step 2: Calculate the Discriminant
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]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]$\Delta = 3^2 - 4(1)(8)$[/tex]
[tex]$\Delta = 9 - 32$[/tex]
[tex]$\Delta = -23$[/tex]
### Step 3: Interpret the Discriminant
The value of the discriminant determines the number and type of roots of the quadratic equation:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], there is one real root (a repeated root).
- If [tex]\(\Delta < 0\)[/tex], there are no real roots; the roots are complex.
Here, [tex]\(\Delta = -23\)[/tex], which is less than 0. This means the quadratic equation [tex]\(x^2 + 3x + 8 = 0\)[/tex] has no real roots.
### Conclusion
The discriminant of the given quadratic equation is [tex]\(\Delta = -23\)[/tex], and it has no real roots.
Thus, the correct answer is:
C. [tex]\(-23\)[/tex] ; no real roots
