Let's proceed step-by-step to determine the correct substitution of the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula from the given quadratic equation.
1. Given Quadratic Equation:
The given quadratic equation is:
[tex]\[
1 = -2x + 3x^2 + 1
\][/tex]
2. Simplifying the Equation:
Subtracting 1 from both sides, we get:
[tex]\[
0 = 3x^2 - 2x + 0
\][/tex]
or:
[tex]\[
3x^2 - 2x = 0
\][/tex]
3. Identifying Coefficients:
We can identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the simplified equation [tex]\(3x^2 - 2x + 0 = 0\)[/tex]:
[tex]\[
a = 3, \quad b = -2, \quad c = 0
\][/tex]
4. Quadratic Formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Substitution into the Quadratic Formula:
Substituting the identified values [tex]\(a = 3\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 0\)[/tex]:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot 0}}{2 \cdot 3}
\][/tex]
Simplifying further:
[tex]\[
x = \frac{2 \pm \sqrt{4 - 0}}{6}
\][/tex]
[tex]\[
x = \frac{2 \pm \sqrt{4}}{6}
\][/tex]
[tex]\[
x = \frac{2 \pm 2}{6}
\][/tex]
6. Solving for [tex]\(x\)[/tex]:
We get two solutions:
[tex]\[
x_1 = \frac{2 + 2}{6} = \frac{4}{6} = \frac{2}{3} \quad \text{or} \quad 0.6666666666666666
\][/tex]
[tex]\[
x_2 = \frac{2 - 2}{6} = \frac{0}{6} = 0
\][/tex]
Thus, the results are [tex]\(x_1 = 0.6666666666666666\)[/tex] (or [tex]\(\frac{2}{3}\)[/tex]) and [tex]\(x_2 = 0\)[/tex].
The expression that shows the correct substitution of the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}
\][/tex]
So, the correct substitution is:
[tex]\[ \boxed{x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}} \][/tex]
