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