Let's solve the quadratic equation [tex]\(x^2 + 2x = -2\)[/tex].
1. Rearrange the equation: Start by bringing all terms to one side to set the equation to zero.
[tex]\[
x^2 + 2x + 2 = 0
\][/tex]
2. Identify the coefficients: This is a standard quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
[tex]\[
a = 1, \quad b = 2, \quad c = 2
\][/tex]
3. Calculate the discriminant: The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (2)^2 - 4(1)(2) = 4 - 8 = -4
\][/tex]
4. Determine the nature of the roots: The discriminant [tex]\(\Delta = -4\)[/tex] is less than zero, indicating that the roots are complex (non-real) and conjugate pairs.
5. Find the roots using the quadratic formula: The quadratic formula for finding roots of [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since [tex]\(\Delta = -4\)[/tex], we can write [tex]\(\sqrt{\Delta} = \sqrt{-4} = 2i\)[/tex]. Therefore:
[tex]\[
x = \frac{-2 \pm 2i}{2(1)} = \frac{-2 \pm 2i}{2} = -1 \pm i
\][/tex]
6. State the solutions: The solutions to the equation [tex]\(x^2 + 2x = -2\)[/tex] are:
[tex]\[
x = -1 - i \quad \text{and} \quad x = -1 + i
\][/tex]
Comparing these solutions with the given options:
- A. [tex]\(x = 1 \pm i\)[/tex]
- B. [tex]\(x = -1 \pm 1\)[/tex]
- C. [tex]\(x = 0\)[/tex]
- D. [tex]\(x = \pm 1\)[/tex]
The correct answer is not explicitly listed in the options you provided. However, if we closely observe, our solutions [tex]\(x = -1 - i\)[/tex] and [tex]\(x = -1 + i\)[/tex] include a complex number [tex]\(i\)[/tex].
Nevertheless, let's carefully conclude the correct answer:
The solution matches the form of [tex]\(-1 \pm i\)[/tex], which unfortunately isn't listed as an explicit option here. So if you provided me complete incorrect set of answers, A would be the closest relevant, but have errors. Given properly correct options, there will be an option explicitly indicating [tex]\(-1 \pm i\)[/tex].
Without any clearly correct selectable option, it seems to be improper information formulating around. Problems could be in choices here.
This leaves need of refurnishment to precise answers listing.
