To solve the quadratic equation [tex]\(-3x^2 - 4x - 4 = 0\)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are as follows:
[tex]\[ a = -3 \][/tex]
[tex]\[ b = -4 \][/tex]
[tex]\[ c = -4 \][/tex]
1. Calculate the Discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-4)^2 - 4(-3)(-4) \][/tex]
[tex]\[ \Delta = 16 - 48 \][/tex]
[tex]\[ \Delta = -32 \][/tex]
2. Determine the solutions:
Since the discriminant is negative ([tex]\(\Delta = -32\)[/tex]), the solutions will involve imaginary numbers.
The quadratic formula then becomes:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\( b \)[/tex], [tex]\( \Delta \)[/tex], and [tex]\( a \)[/tex]:
[tex]\[ x = \frac{-(-4) \pm \sqrt{-32}}{2(-3)} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{-32}}{-6} \][/tex]
3. Simplify the square root of the negative discriminant:
Recall that [tex]\( \sqrt{-1} = i \)[/tex]:
[tex]\[ \sqrt{-32} = \sqrt{-1 \cdot 32} \][/tex]
[tex]\[ \sqrt{-32} = \sqrt{-1} \cdot \sqrt{32} \][/tex]
[tex]\[ \sqrt{-32} = i \cdot \sqrt{32} \][/tex]
[tex]\[ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \][/tex]
Therefore:
[tex]\[ \sqrt{-32} = 4i\sqrt{2} \][/tex]
4. Insert back into the formula and simplify:
[tex]\[ x = \frac{4 \pm 4i\sqrt{2}}{-6} \][/tex]
Simplify by dividing both the numerator and the denominator by 2:
[tex]\[ x = \frac{2 \pm 2i\sqrt{2}}{-3} \][/tex]
Since the denominator is -3, distribute the negative sign:
[tex]\[ x = \frac{-2 \mp 2i\sqrt{2}}{3} \][/tex]
Finally, the solutions to the quadratic equation [tex]\( -3x^2 - 4x - 4 = 0 \)[/tex] are:
[tex]\[ x = \frac{-2 \pm 2i\sqrt{2}}{3} \][/tex]
Among the given options:
[tex]\[ x=\frac{-2 \pm 2 i \sqrt{2}}{3} \][/tex]
This is correct.
