Let's begin with the given quadratic equation:
[tex]\[ 0 = -3x^2 - 2x + 6 \][/tex]
We need to substitute the values [tex]\( a = -3 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 6 \)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's break down each component of the quadratic formula step-by-step:
1. Substitute [tex]\( b \)[/tex]:
[tex]\[ -(-2) = 2 \][/tex]
2. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[ (-2)^2 = 4 \][/tex]
3. Calculate [tex]\( 4ac \)[/tex]:
[tex]\[ 4 \cdot (-3) \cdot 6 = 4 \cdot -18 = -72 \][/tex]
4. Calculate the discriminant ([tex]\( b^2 - 4ac \)[/tex]):
[tex]\[ 4 - (-72) = 4 + 72 = 76 \][/tex]
5. Calculate [tex]\( 2a \)[/tex]:
[tex]\[ 2 \cdot (-3) = -6 \][/tex]
Thus, the correct values substituted into the quadratic formula from the given equation, taking [tex]\( a = -3 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 6 \)[/tex], are:
[tex]\[ x = \frac{2 \pm \sqrt{76}}{-6} \][/tex]
