To solve the quadratic equation \( x^2 + 32x + 248 = 0 \), we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, the coefficients are:
- \( a = 1 \)
- \( b = 32 \)
- \( c = 248 \)
Let's calculate the discriminant (\( b^2 - 4ac \)) first:
\[ b^2 = 32^2 = 1024 \] \[ 4ac = 4 \times 1 \times 248 = 992 \] \[ b^2 - 4ac = 1024 - 992 = 32 \]
Now, substituting the values into the quadratic formula:
\[ x = \frac{-32 \pm \sqrt{32}}{2 \times 1} \]
Calculating \( \sqrt{32} \):
\[ \sqrt{32} = 4\sqrt{2} \]
Now substitute back into the formula:
\[ x = \frac{-32 \pm 4\sqrt{2}}{2} \]
Simplifying further:
\[ x = \frac{-32}{2} \pm \frac{4\sqrt{2}}{2} \] \[ x = -16 \pm 2\sqrt{2} \]
Thus, the solutions to the equation \( x^2 + 32x + 248 = 0 \) are:
\[ x = -16 + 2\sqrt{2} \quad \text{and} \quad x = -16 - 2\sqrt{2} \]
So none of the options presented seem to be correct in this form. The solutions can also be approximately calculated as:
\[ x \approx -16 + 2(1.414) \quad \text{and} \quad x \approx -16 - 2(1.414) \] This gives us approximately:
- \( x \approx -13.172 \)
- \( x \approx -18.828 \)
Please confirm if you need further clarification!