To determine which of the given options are solutions to the quadratic equation [tex]\(0 = 2x^2 + x + 4\)[/tex], we need to solve the equation [tex]\(2x^2 + x + 4 = 0\)[/tex].
### Step 1: Understand Solutions of Quadratic Equations
For any quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], the solutions are given by the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 2: Identify the Coefficients
In this given quadratic equation [tex]\(2x^2 + x + 4 = 0\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 1\)[/tex]
- [tex]\(c = 4\)[/tex]
### Step 3: Compute the Discriminant
The discriminant [tex]\(\Delta\)[/tex] is calculated as:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 1^2 - 4(2)(4) \][/tex]
[tex]\[ \Delta = 1 - 32 \][/tex]
[tex]\[ \Delta = -31 \][/tex]
Since the discriminant is negative, we will have complex roots.
### Step 4: Apply the Quadratic Formula
Using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{-31}}{2(2)} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{31}i}{4} \][/tex]
Thus, the solutions to the quadratic equation are:
[tex]\[ x = \frac{-1 - \sqrt{31}i}{4} \][/tex]
[tex]\[ x = \frac{-1 + \sqrt{31}i}{4} \][/tex]
### Step 5: Match the Solutions with the Options
Now we compare the provided options with our solutions:
Option A: [tex]\(\frac{-1 - i \sqrt{23}}{2}\)[/tex]
- This does not match either of our solutions.
Option B: [tex]\(\frac{-1 + i \sqrt{10}}{4}\)[/tex]
- This does not match either of our solutions.
Option C: [tex]\(\frac{-1 - i \sqrt{31}}{4}\)[/tex]
- This matches one of our solutions: [tex]\( x = \frac{-1 - \sqrt{31}i}{4} \)[/tex].
Option D: [tex]\(\frac{-1 - 3 i}{2}\)[/tex]
- This does not match either of our solutions.
### Conclusion
Among the given options, the correct solution to the quadratic equation [tex]\(2x^2 + x + 4 = 0\)[/tex] is:
[tex]\[ \boxed{\frac{-1 - i \sqrt{31}}{4}} \][/tex]
