Question

Solve −3x2 − 4x − 4 = 0.

x equals quantity of 2 plus or minus 4i square root of 2 all over 3
x equals quantity of 2 plus or minus 2i square root of 2 all over 3
x equals quantity of negative 2 plus or minus 2i square root of 2 all over 3
x equals quantity of negative 2 plus or minus 4i square root of 2 all over 3

Answer 1

To solve the quadratic equation \(-3x^2 - 4x - 4 = 0\), we will use the quadratic formula:

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

where \(a = -3\), \(b = -4\), and \(c = -4\).

1. **Calculate the discriminant** \(b^2 - 4ac\):

\[
b^2 = (-4)^2 = 16
\]

\[
4ac = 4 \cdot (-3) \cdot (-4) = 48
\]

\[
b^2 - 4ac = 16 - 48 = -32
\]

Since the discriminant is negative, we will have complex solutions.

2. **Calculate \(x\) using the quadratic formula**:

\[
x = \frac{-(-4) \pm \sqrt{-32}}{2 \cdot -3} = \frac{4 \pm \sqrt{-32}}{-6}
\]

3. **Simplify \(\sqrt{-32}\)**:

\[
\sqrt{-32} = \sqrt{32} \cdot i = \sqrt{16 \cdot 2} \cdot i = 4\sqrt{2}i
\]

4. **Substituting back:**

\[
x = \frac{4 \pm 4\sqrt{2}i}{-6}
\]

5. **Further simplify the expression**:

\[
x = \frac{4}{-6} \pm \frac{4\sqrt{2}i}{-6}
\]

\[
x = -\frac{2}{3} \pm \frac{2\sqrt{2}}{3}i
\]

Thus, the solutions can be written as:

\[
x = -\frac{2}{3} \pm \frac{2\sqrt{2}}{3} i
\]

This is equivalent to:

\[
x = -2/3 \pm 2i\sqrt{2}/3
\]

Now we compare it with the options given:

- \(x\) equals quantity of \(-2\) plus or minus \(2i \sqrt{2}\) all over \(3\).

Thus the correct answer is:

**x equals quantity of negative 2 plus or minus 2i square root of 2 all over 3**.