To find the x-intercepts of the quadratic equation \( y = x^2 + 3x + 8 \), we set \( y = 0 \) and solve for \( x \):
\[ 0 = x^2 + 3x + 8 \]
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 1 \), \( b = 3 \), and \( c = 8 \). Plugging these values into the formula:
- Calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 8 = 9 - 32 = -23 \]
Since the discriminant is negative, this means the quadratic has no real roots. Therefore, there are no x-intercepts.
Thus, the correct response is:
There is no x-intercept.