We are tasked with solving the quadratic equation:

$x^2 + 2x + 17 = 0$

To solve this, we can apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $x^2 + 2x + 17 = 0$, we identify the coefficients:

• $a = 1$
• $b = 2$
• $c = 17$

Now, let's compute the discriminant:

$\Delta = b^2 - 4ac = (2)^2 - 4(1)(17) = 4 - 68 = -64$

Since the discriminant is negative ($\Delta = -64$), the solutions will be complex. We proceed as follows:

$x = \frac{-2 \pm \sqrt{-64}}{2(1)} = \frac{-2 \pm 8i}{2}$

This simplifies to:

$x = \frac{-2}{2} \pm \frac{8i}{2} = -1 \pm 4i$

Therefore, the solutions are:

$x = -1 + 4i \quad \text{or} \quad x = -1 - 4i$

The correct answer is option A.

Would you like further details or have any questions?

Here are some related questions for practice:

1. Solve $x^2 + 4x + 13 = 0$ using the quadratic formula.
2. Explain how the discriminant affects the nature of roots in a quadratic equation.
3. What are the steps to simplify complex numbers when solving quadratics?
4. Find the roots of $x^2 - 6x + 25 = 0$.
5. Can a quadratic equation with real coefficients have only one complex root? Why or why not?

Tip: The discriminant ($b^2 - 4ac$) tells you whether the roots of a quadratic are real, repeated, or complex.