Let's solve the problem step by step using the quadratic formula:
The equation given is [tex]\(0 = -3x^2 - 2x + 6\)[/tex], which means our values for the quadratic formula are:
[tex]\[ a = -3, \quad b = -2, \quad c = 6 \][/tex]
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Now let's substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula.
1. Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-2)^2 - 4(-3)(6) \][/tex]
2. Compute [tex]\(b^2\)[/tex]:
[tex]\[ (-2)^2 = 4 \][/tex]
3. Compute [tex]\(4ac\)[/tex]:
[tex]\[ 4(-3)(6) = 4 \cdot -3 \cdot 6 = -72 \][/tex]
4. Compute the discriminant:
[tex]\[ 4 - (-72) = 4 + 72 = 76 \][/tex]
5. Substitute [tex]\(b\)[/tex] and the discriminant back into the formula:
[tex]\[ x = \frac{-(-2) \pm \sqrt{76}}{2(-3)} \][/tex]
6. Simplify [tex]\(-b\)[/tex]:
[tex]\[ -(-2) = 2 \][/tex]
Thus,
[tex]\[ x = \frac{2 \pm \sqrt{76}}{-6} \][/tex]
Hence, the correct substitution into the quadratic formula should be:
[tex]\[ x = \frac{2 \pm \sqrt{76}}{-6} \][/tex]
Given the numerical values:
[tex]\[ \sqrt{76} \approx 8.7178 \][/tex]
So we have two solutions:
[tex]\[ x_1 = \frac{2 + 8.7178}{-6} \approx \frac{10.7178}{-6} \approx -1.7863 \][/tex]
[tex]\[ x_2 = \frac{2 - 8.7178}{-6} \approx \frac{-6.7178}{-6} \approx 1.1196 \][/tex]
Therefore, the correct full step-by-step substitutions and results are:
[tex]\[ x = \frac{10.7178}{-6}, \quad x = \frac{-6.7178}{-6}. \][/tex]
Thus, the expressions for these substitutions are:
[tex]\[ x = \frac{10.717797887081348}{-6} \quad \text{and} \quad x = \frac{-6.717797887081348}{-6} \][/tex]
These represent the correct substitution and results for the given values [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex].
