To solve the equation [tex]\((x + 1)^2 - 2 = \frac{2}{2}\)[/tex] step-by-step, follow these procedures:
1. Simplify the Right Side of the Equation:
[tex]\[
\frac{2}{2} = 1
\][/tex]
So, the equation becomes:
[tex]\[
(x + 1)^2 - 2 = 1
\][/tex]
2. Isolate the Squared Term:
Add 2 to both sides of the equation:
[tex]\[
(x + 1)^2 - 2 + 2 = 1 + 2
\][/tex]
Simplifying this, we get:
[tex]\[
(x + 1)^2 = 3
\][/tex]
3. Solve for [tex]\(x + 1\)[/tex]:
Take the square root of both sides of the equation:
[tex]\[
x + 1 = \pm \sqrt{3}
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
Subtract 1 from both possible solutions of [tex]\(x + 1\)[/tex]:
[tex]\[
x + 1 = \sqrt{3} \implies x = \sqrt{3} - 1
\][/tex]
[tex]\[
x + 1 = -\sqrt{3} \implies x = -\sqrt{3} - 1
\][/tex]
So, the two solutions for [tex]\(x\)[/tex] are:
[tex]\[
x = \sqrt{3} - 1 \quad \text{and} \quad x = -\sqrt{3} - 1
\][/tex]
5. Approximate the Solutions:
Using the approximate value of [tex]\(\sqrt{3} \approx 1.732\)[/tex]:
[tex]\[
x \approx 1.732 - 1 \approx 0.732
\][/tex]
[tex]\[
x \approx -1.732 - 1 \approx -2.732
\][/tex]
Given the equations and approximations, we can match the solutions with the options in the table. The solutions approximately are:
[tex]\[
x \approx 0.732 \quad \text{and} \quad x \approx -2.732
\][/tex]
Thus, from the table, neither [tex]\(x = 2\)[/tex], [tex]\(x = 0.5\)[/tex], [tex]\(x = 1\)[/tex], [tex]\(x = 0.25\)[/tex], [tex]\(x = -2\)[/tex], [tex]\(x = 0\)[/tex], nor [tex]\(x = -0.5\)[/tex] match our solutions. There are no direct matches in the provided table with our solved values.
The correct approximate solutions based on calculations are:
[tex]\[
x \approx -2.732 \quad \text{and} \quad x \approx 0.732
\][/tex]
