Let's solve the quadratic equation $x^2 + 2x + 3 = 0$ and determine the correct solution.

Step 1: Use the Quadratic Formula

The quadratic formula is:

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

For the equation $x^2 + 2x + 3 = 0$:

• $a = 1$
• $b = 2$
• $c = 3$

Step 2: Calculate the Discriminant

$b^2 - 4ac = (2)^2 - 4(1)(3) = 4 - 12 = -8$

Since the discriminant is negative, the solutions involve imaginary numbers.

Step 3: Compute the Square Root

$\sqrt{-8} = \sqrt{8} i = 2\sqrt{2} i$

Step 4: Compute the Two Solutions

$x = \frac{-2 \pm 2\sqrt{2} i}{2}$

$x = -1 \pm i\sqrt{2}$

Step 5: Identify the Correct Option

Comparing with the given options:

• Option 1: $x = -1 \pm i\sqrt{2}$ ✅ (Correct)
• Option 2: $x = -1 \pm 2i$ ❌ (Incorrect)
• Option 3: $x = 1 \pm i\sqrt{2}$ ❌ (Incorrect)

Final Answer:

The correct choice is Option 1.

Would you like a detailed explanation of any step?

Here are five related questions to explore further:

1. How do we determine the nature of the roots using the discriminant?
2. What are the implications of a negative discriminant in a quadratic equation?
3. How can we verify solutions by substituting them back into the equation?
4. What is the geometric interpretation of complex roots in quadratic equations?
5. How does the quadratic formula derive from completing the square?

Tip: Always check the discriminant before solving a quadratic equation to determine whether the roots are real or complex.