To solve the given quadratic equation [tex]\(x^2 + 1 = 2x - 3\)[/tex], we first need to rearrange it into standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
Starting with the equation:
[tex]\[
x^2 + 1 = 2x - 3
\][/tex]
We move all terms to one side to set the equation to zero:
[tex]\[
x^2 + 1 - 2x + 3 = 0
\][/tex]
This simplifies to:
[tex]\[
x^2 - 2x + 4 = 0
\][/tex]
In the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], we identify the coefficients [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 4\)[/tex].
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substituting the coefficients [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 4\)[/tex] into the quadratic formula, we get:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}
\][/tex]
Now, let's examine each of the provided expressions to determine which one matches our result:
A. [tex]\(\frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}\)[/tex]
B. [tex]\(\frac{-(-2) \pm \sqrt{(-2)^2 - (1)(4)}}{2(2)}\)[/tex]
C. [tex]\(\frac{-2 \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}\)[/tex]
D. [tex]\(\frac{-2 \pm \sqrt{(2)^2 - 4(1)(-2)}}{2(1)}\)[/tex]
Evaluating each option:
- Option A correctly matches the form that we derived from the quadratic formula.
- Option B has an incorrect coefficient for [tex]\(c\)[/tex] in the discriminant and incorrect denominator.
- Option C has an incorrect sign for the numerator term [tex]\(-b\)[/tex].
- Option D has incorrect values in the discriminant and [tex]\(-b\)[/tex] terms.
Therefore, the expression that correctly sets up the quadratic formula for the equation [tex]\(x^2 - 2x + 4 = 0\)[/tex] is:
[tex]\[
\boxed{\frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}}
\][/tex]
Hence, the correct expression is option A.
