To solve the quadratic equation [tex]\(x^2 + 1 = 2x - 3\)[/tex], we need to first bring all terms to one side of the equation to have it in standard form:
[tex]\[x^2 + 1 - 2x + 3 = 0.\][/tex]
Simplifying the equation:
[tex]\[x^2 - 2x + 4 = 0.\][/tex]
This is a standard quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] where [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]
We need to find the correct expression that sets up the quadratic formula. Let's analyze each option:
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]
Step-by-step Analysis:
1. Substitute [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 4\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}. \][/tex]
2. Simplify within the discriminant (under the square root):
[tex]\[
(-2)^2 = 4,
\][/tex]
[tex]\[
4ac = 4 \times 1 \times 4 = 16,
\][/tex]
[tex]\[
b^2 - 4ac = 4 - 16 = -12.
\][/tex]
3. Substitute back into the formula:
[tex]\[ x = \frac{2 \pm \sqrt{-12}}{2}. \][/tex]
Confirming Results:
- Option A:
[tex]\[ \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)} \][/tex]
[tex]\[ \frac{2 \pm \sqrt{4 - 16}}{2} = \frac{2 \pm \sqrt{-12}}{2} \][/tex]
This matches our calculation.
- Option B:
[tex]\[ \frac{-(-2) \pm \sqrt{(-2)^2 - (1)(4)}}{2(2)} \][/tex]
This has an incorrect explanation within the discriminant [tex]\((-2)^2 - (1)(4) = 4 - 4 = 0\)[/tex] and wrong denominator [tex]\(2(2)\)[/tex].
- Option C:
[tex]\[ \frac{-2 \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)} \][/tex]
This does not match correctly as it uses [tex]\( -2 \)[/tex] instead of [tex]\( 2 \)[/tex].
- Option D:
[tex]\[ \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-2)}}{2(1)} \][/tex]
This contains [tex]\( -4 \)[/tex].\ ([tex]\(2)^2 - 4(1)(-2) = 4 + 8 = 12\)[/tex]
Thus, the correct expression is:
[tex]\(\boxed{\frac{2 \pm \sqrt{-12}}{2}}\)[/tex]
Therefore, the correct expression that sets up the quadratic formula is:
[tex]\[ \boxed{A. \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}} \][/tex]
Thus, the correct answer is [tex]\(1\)[/tex].
