Let's solve this step-by-step and find which option shows the correct substitution of [tex]\(a, b\)[/tex], and [tex]\(c\)[/tex] from the quadratic equation [tex]\(0 = -3x^2 - 2x + 6\)[/tex] into the quadratic formula.
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Given quadratic equation:
[tex]\[ 0 = -3x^2 - 2x + 6 \][/tex]
From this equation, we can identify the coefficients:
[tex]\[ a = -3 \][/tex]
[tex]\[ b = -2 \][/tex]
[tex]\[ c = 6 \][/tex]
Substitute [tex]\(a, b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
So, let's compare this with the available choices:
1. [tex]\( x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \)[/tex]
2. [tex]\( x = \frac{-2 \pm \sqrt{2^2 - 4(-3)(6)}}{2(-3)} \)[/tex]
3. [tex]\( x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(6)}}{2(3)} \)[/tex]
4. [tex]\( x = \frac{-2 \pm \sqrt{2^2 - 4(3)(6)}}{2(3)} \)[/tex]
Choice 1 matches the correct substitution:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
Therefore, the correct answer is:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
This shows the proper substitution of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the given quadratic equation into the quadratic formula.
