Let's analyze the equation [tex]\( 1 = -2x + 3x^2 + 1 \)[/tex] and find the correct substitution of the coefficients [tex]\( a, b,\)[/tex] and [tex]\( c \)[/tex] into the quadratic formula.
1. First, rewrite the equation to standard quadratic form:
The given equation is [tex]\( 1 = -2x + 3x^2 + 1 \)[/tex].
Simplify the equation by subtracting 1 from both sides:
[tex]\[
0 = -2x + 3x^2
\][/tex]
Rewriting it in the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex],
[tex]\[
3x^2 - 2x + 0 = 0
\][/tex]
Here, we identify the coefficients:
[tex]\[
a = 3, \quad b = -2, \quad c = 0
\][/tex]
2. Substitute [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] into the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute the identified values [tex]\( a = 3 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 0 \)[/tex]:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}
\][/tex]
Hence, the correct substitution is:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}
\][/tex]
