Question

Determine which of these potential solutions make the equation x2+2x+3=0 true. Option 1: x=−1±i2–√ Option 2: x=−1±2i Option 3: x=1±i2–√

Answer 1

To determine which of the potential solutions make the equation \( x^2 + 2x + 3 = 0 \) true, we will test each option by substituting the values into the equation.

First, let's find the roots of the equation \( x^2 + 2x + 3 = 0 \) using the quadratic formula:

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

where \( a = 1 \), \( b = 2 \), and \( c = 3 \).

Calculating the discriminant:

\[
b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8
\]

Since the discriminant is negative, the roots will be complex numbers. Now, substituting into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{-8}}{2 \cdot 1} = \frac{-2 \pm 2i\sqrt{2}}{2} = -1 \pm i\sqrt{2}
\]

So, the solutions to the equation are given by:

\[
x = -1 \pm i\sqrt{2}
\]

Now we will analyze each option:

1. **Option 1: \( x = -1 \pm i\sqrt{2} \):**
- This matches with our derived solutions. So this option is a solution.

2. **Option 2: \( x = -1 \pm 2i \):**
- The solutions here are not the same as \( -1 \pm i\sqrt{2} \). Hence, this option is not a solution.

3. **Option 3: \( x = 1 \pm i\sqrt{2} \):**
- The solutions here are also not the same as \( -1 \pm i\sqrt{2} \). Hence, this option is not a solution.

Thus, the only option that makes the equation \( x^2 + 2x + 3 = 0 \) true is:

**Option 1: \( x = -1 \pm i\sqrt{2} \)**.