To find the x-intercepts of the quadratic function \( y = x^2 + 3x + 8 \), we need to set \( y \) equal to 0 and solve for \( x \):
\[ 0 = x^2 + 3x + 8 \]
Next, we can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 3 \), and \( c = 8 \):
- Calculate the discriminant \( b^2 - 4ac \):
\[ 3^2 - 4 \cdot 1 \cdot 8 = 9 - 32 = -23 \]
Since the discriminant is negative, this means that there are no real x-intercepts for this quadratic function.
Now let's look at the provided responses:
- (-1.5, 5.75)
- There is no x-intercept. (This one is correct)
- (0, 8)
- -1.5
The correct answer is There is no x-intercept.