First, identify the coefficients a, b, and c in the quadratic equation ax^2 + bx + c = 0. In this case, we will say:
a = 1
b = 3
c = 8
Second, find the discriminant. In a quadratic equation, the discriminant (D) can be found using the formula D = b^2 - 4ac.
In this case, our discriminant will be:
D = b^2 -4ac
D = (3)^2 - 4 * (1) * (8)
D = 9 - 32
D = -23
So, the discriminant for the equation x^2 + 3x + 8 = 0 is -23.
Finally, we can use the discriminant to determine the number of real roots the equation has.
The value of the discriminant tells us about the roots of a quadratic equation:
- If the discriminant is greater than zero, the equation has two distinct real roots.
- If the discriminant is equal to zero, the equation has one real root (or a repeated root).
- If the discriminant is less than zero, the equation has no real roots.
In this case, our discriminant is -23, which is less than zero. Therefore, the equation x^2 + 3x + 8 = 0 has no real roots.
In summary, the discriminant of x^2 + 3x + 8 = 0 is -23 and this equation has no real roots.