To solve the quadratic equation [tex]\(-3x^2 - 4x - 4 = 0\)[/tex], let's proceed with the steps to find the roots.
### Step 1: Identify the coefficients
From the quadratic equation in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], we have:
[tex]\[ a = -3 \][/tex]
[tex]\[ b = -4 \][/tex]
[tex]\[ c = -4 \][/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\(D\)[/tex] is given by:
[tex]\[ D = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ D = (-4)^2 - 4(-3)(-4) \][/tex]
[tex]\[ D = 16 - 48 \][/tex]
[tex]\[ D = -32 \][/tex]
Since the discriminant [tex]\(D\)[/tex] is negative ([tex]\(D = -32\)[/tex]), the roots will be complex numbers.
### Step 3: Use the quadratic formula
The quadratic formula for solving [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{D}}{2a} \][/tex]
For complex solutions, we use:
[tex]\[ x = \frac{-b \pm i\sqrt{|D|}}{2a} \][/tex]
First, we find [tex]\(\sqrt{|D|}\)[/tex]:
[tex]\[ \sqrt{|D|} = \sqrt{32} = 4\sqrt{2} \][/tex]
Next, we use the quadratic formula:
[tex]\[ x = \frac{-(-4) \pm i \cdot 4\sqrt{2}}{2(-3)} \][/tex]
[tex]\[ x = \frac{4 \pm i \cdot 4\sqrt{2}}{-6} \][/tex]
[tex]\[ x = \frac{4}{-6} \pm \frac{i \cdot 4\sqrt{2}}{-6} \][/tex]
[tex]\[ x = -\frac{2}{3} \pm \frac{2i\sqrt{2}}{3} \][/tex]
So, the solutions are:
[tex]\[ x = -\frac{2}{3} + \frac{2i\sqrt{2}}{3} \quad \text{and} \quad x = -\frac{2}{3} - \frac{2i\sqrt{2}}{3} \][/tex]
### Step 4: Verify with the given solutions
We are given the solutions to check:
1. [tex]\( x = \frac{2 \pm 4i\sqrt{2}}{3} \)[/tex]
2. [tex]\( x = \frac{2 \pm 2i\sqrt{2}}{3} \)[/tex]
3. [tex]\( x = -\frac{2 \pm 2i\sqrt{2}}{3} \)[/tex]
4. [tex]\( x = -\frac{2 \pm 4i\sqrt{2}}{3} \)[/tex]
Our calculated solutions are:
[tex]\[ x = -\frac{2}{3} + \frac{2i\sqrt{2}}{3} \][/tex]
[tex]\[ x = -\frac{2}{3} - \frac{2i\sqrt{2}}{3} \][/tex]
Comparing with the given options, we see that the correct solutions matching our results are:
[tex]\[ \boxed{\frac{-2 \pm 2i\sqrt{2}}{3}} \][/tex]
