We need to determine which of the given quadratic equations has the solutions [tex]\(x = \frac{-3 \pm \sqrt{3}i}{2}\)[/tex].
Given Equations:
1. [tex]\(2x^2 + 6x + 9 = 0\)[/tex]
2. [tex]\(x^2 + 3x + 12 = 0\)[/tex]
3. [tex]\(x^2 + 3x + 3 = 0\)[/tex]
4. [tex]\(2x^2 + 6x + 3 = 0\)[/tex]
We'll compare the given solutions [tex]\(x = \frac{-3 \pm \sqrt{3}i}{2}\)[/tex] with the solutions of each equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex].
Step-by-step Solution:
### 1. Solve [tex]\(2x^2 + 6x + 9 = 0\)[/tex]
Using the quadratic formula:
[tex]\[ a = 2, \, b = 6, \, c = 9 \][/tex]
[tex]\[ x = \frac{-6 \pm \sqrt{6^2 - 4(2)(9)}}{2(2)} \][/tex]
[tex]\[ x = \frac{-6 \pm \sqrt{36 - 72}}{4} \][/tex]
[tex]\[ x = \frac{-6 \pm \sqrt{-36}}{4} \][/tex]
[tex]\[ x = \frac{-6 \pm 6i}{4} \][/tex]
[tex]\[ x = \frac{-6 \pm 6i}{4} \][/tex]
[tex]\[ x = \frac{-3 \pm 3i}{2} \][/tex]
This does not match the given solutions.
### 2. Solve [tex]\(x^2 + 3x + 12 = 0\)[/tex]
Using the quadratic formula:
[tex]\[ a = 1, \, b = 3, \, c = 12 \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4(1)(12)}}{2(1)} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{9 - 48}}{2} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{-39}}{2} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{39}i}{2} \][/tex]
This does not match the given solutions.
### 3. Solve [tex]\(x^2 + 3x + 3 = 0\)[/tex]
Using the quadratic formula:
[tex]\[ a = 1, \, b = 3, \, c = 3 \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4(1)(3)}}{2(1)} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{9 - 12}}{2} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{-3}}{2} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{3}i}{2} \][/tex]
This matches the given solutions [tex]\(x = \frac{-3 \pm \sqrt{3}i}{2}\)[/tex].
### 4. Solve [tex]\(2x^2 + 6x + 3 = 0\)[/tex]
Using the quadratic formula:
[tex]\[ a = 2, \, b = 6, \, c = 3 \][/tex]
[tex]\[ x = \frac{-6 \pm \sqrt{6^2 - 4(2)(3)}}{2(2)} \][/tex]
[tex]\[ x = \frac{-6 \pm \sqrt{36 - 24}}{4} \][/tex]
[tex]\[ x = \frac{-6 \pm \sqrt{12}}{4} \][/tex]
[tex]\[ x = \frac{-6 \pm 2\sqrt{3}}{4} \][/tex]
[tex]\[ x = \frac{-3 \pm \sqrt{3}}{2} \][/tex]
This does not match the given solutions.
Therefore, the equation that has the solutions [tex]\(x = \frac{-3 \pm \sqrt{3}i}{2}\)[/tex] is:
[tex]\[ x^2 + 3x + 3 = 0 \][/tex]
So, the correct option is:
[tex]\[ x^2 + 3x + 3 = 0 \][/tex]
