To determine the x-intercepts of the quadratic function \( y = x^2 + 3x + 8 \), we need to find the values of \( x \) for which \( y = 0 \). This means solving the equation:
\[ x^2 + 3x + 8 = 0 \]
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, \( a = 1 \), \( b = 3 \), and \( c = 8 \). Plugging these values into the formula gives:
\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \]
Calculating the discriminant:
\[ b^2 - 4ac = 9 - 32 = -23 \]
Since the discriminant is negative (\(-23\)), this indicates that there are no real x-intercepts.
Thus, the answer to the question of how many x-intercepts there are is:
A. there is on x-intercept (which, in fact, is incorrect: there are actually no x-intercepts at all).
To clarify, the function does not cross the x-axis because the parabola opens upwards (since the coefficient of \(x^2\) is positive) and the vertex is above the x-axis (as indicated by the negative discriminant).
So, the correct answer is that there are no x-intercepts.