Let's start by looking at the equation given: [tex]\(1 = -2x + 3x^2 + 1\)[/tex]. We can rewrite this equation in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
First, subtract 1 from both sides of the equation:
[tex]\[1 - 1 = -2x + 3x^2 + 1 - 1\][/tex]
[tex]\[0 = -2x + 3x^2\][/tex]
Which simplifies to:
[tex]\[3x^2 - 2x = 0\][/tex]
Here, the quadratic equation we have is:
[tex]\[3x^2 - 2x = 0\][/tex]
In the standard form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[a = 3\][/tex]
[tex]\[b = -2\][/tex]
[tex]\[c = 0\][/tex]
The quadratic formula is:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
Substitute the values [tex]\(a = 3\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 0\)[/tex] into the quadratic formula:
[tex]\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}\][/tex]
Simplify the expression step by step:
1. Calculate the term [tex]\(-b\)[/tex]:
[tex]\[ -(-2) = 2\][/tex]
2. Calculate the term [tex]\(b^2\)[/tex]:
[tex]\[(-2)^2 = 4\][/tex]
3. Calculate the discriminant ([tex]\(\Delta\)[/tex]), which is the term [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[\Delta = 4 - 4(3)(0)\][/tex]
[tex]\[\Delta = 4 - 0\][/tex]
[tex]\[\Delta = 4\][/tex]
So, the correct substitution into the quadratic formula is:
[tex]\[x = \frac{2 \pm \sqrt{4}}{6}\][/tex]
[tex]\[x = \frac{2 \pm 2}{6}\][/tex]
Thus, the correct substitution of the values [tex]\(a, b\)[/tex], and [tex]\(c\)[/tex] from the equation into the quadratic formula is:
[tex]\[x = \frac{2 \pm \sqrt{4 - 12}}{6}\][/tex]
