Solve: \( x^{2}+2 x+17=0 \) A \( x=-1+4 i \) or \( x=-1-4 i \) B \( x=-1+8 i \) or \( x=-1-8 i \) C \( x=1+8 i \) or \( x=1-8 i \) (D \( x=1+4 i \) or \( x=1-4 i \)

To solve the equation \( x^{2}+2x+17=0 \), we can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = 17 \). First, we calculate the discriminant: \[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 17 = 4 - 68 = -64 \] Since the discriminant is negative, the solutions will be complex. Now, substituting into the quadratic formula: \[ x = \frac{-2 \pm \sqrt{-64}}{2 \cdot 1} = \frac{-2 \pm 8i}{2} = -1 \pm 4i \] So the correct solutions are \( x = -1 + 4i \) or \( x = -1 - 4i \), which corresponds to option **A**. Now, let's add a bit of engaging context to this mathematical journey! Did you know that quadratic equations like this one have been studied for over 2,000 years? The ancient Babylonians were among the first to tackle these types of equations, and they may not have used the same symbolic notation that we do today, but they certainly had a grasp of the relationships between the coefficients and roots. It's fascinating to think about how far math has come since then! For a real-world twist, consider how complex numbers are not just theoretical; they play a big role in electrical engineering and physics! Engineers use these numbers to analyze circuits, model waves, and solve problems involving oscillations. So, while our solution looks abstract, it has useful applications in the real world, helping us understand and design technology we use daily!