Solving (x+1)^2-2=2/x Algebraically With Solution Selection

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Step-by-Step Solution

To solve the equation $(x+1)^2 - 2 = \frac{2}{x}$ algebraically, we'll follow these steps:

1. Eliminate the Fraction:

The first crucial step in solving the equation is to eliminate the fraction to simplify the equation. Multiply both sides of the equation by $x$ to get rid of the denominator:

$x[(x+1)^2 - 2] = x(\frac{2}{x})$

This simplifies to:

$x(x+1)^2 - 2x = 2$

2. Expand and Simplify:

Next, we expand the squared term and simplify the equation. Expand $(x+1)^2$:

$(x+1)^2 = x^2 + 2x + 1$

Substitute this back into the equation:

$x(x^2 + 2x + 1) - 2x = 2$

Distribute the $x$:

$x^3 + 2x^2 + x - 2x = 2$

Combine like terms:

$x^3 + 2x^2 - x = 2$

3. Rearrange the Equation:

To solve for $x$, we rearrange the equation into a standard polynomial form by setting it equal to zero:

$x^3 + 2x^2 - x - 2 = 0$

4. Factor the Polynomial:

Now, we factor the cubic polynomial. We can try factoring by grouping:

$(x^3 + 2x^2) + (-x - 2) = 0$

Factor out the greatest common factor from each group:

$x^2(x + 2) - 1(x + 2) = 0$

Notice that $(x + 2)$ is a common factor:

$(x + 2)(x^2 - 1) = 0$

Further factor the difference of squares $(x^2 - 1)$:

$(x + 2)(x - 1)(x + 1) = 0$

5. Solve for x:

To find the solutions, we set each factor equal to zero:

• $x + 2 = 0 \Rightarrow x = -2$
• $x - 1 = 0 \Rightarrow x = 1$
• $x + 1 = 0 \Rightarrow x = -1$

Thus, the solutions are $x = -2$, $x = 1$, and $x = -1$.

Identifying Solutions in the Table

From the given solution options:

• $x = -1$
• $x = 0$
• $x = 2$

We compare these with our solutions $x = -2$, $x = 1$, and $x = -1$. The solution $x = -1$ is present in the options.

Let's check if these solutions satisfy the original equation $(x+1)^2 - 2 = \frac{2}{x}$:

1. For $x = -1$:

((-1) + 1)^2 - 2 = \frac{2}{-1}

(0)^2 - 2 = -2

-2 = -2 (True)

So, $x = -1$ is a valid solution.

2. For $x = 0$:

Since we have $\frac{2}{x}$ in the original equation, $x$ cannot be 0 because division by zero is undefined. Thus, $x = 0$ is not a solution.

3. For $x = 2$:

(2 + 1)^2 - 2 = \frac{2}{2}

(3)^2 - 2 = 1

9 - 2 = 1

7 = 1 (False)

So, $x = 2$ is not a solution.

Therefore, from the given options, only $x = -1$ is a solution.

Conclusion

In conclusion, we have solved the equation $(x+1)^2 - 2 = \frac{2}{x}$ algebraically and identified the solutions. By eliminating the fraction, expanding and simplifying, rearranging into a polynomial, factoring, and solving for $x$, we found the solutions to be $x = -2$, $x = 1$, and $x = -1$. Comparing these with the given options, we determined that only $x = -1$ is a valid solution from the provided table. This exercise demonstrates the importance of careful algebraic manipulation and verification of solutions in the original equation. Understanding these steps is crucial for solving similar algebraic problems accurately. Mastering these techniques empowers you to tackle complex equations with confidence and precision, reinforcing your algebraic skills and enhancing your problem-solving capabilities. Remember to always check your solutions in the original equation to ensure they are valid, and practice these steps to build your proficiency in algebra. By consistently applying these methods, you will develop a strong foundation in solving algebraic equations, a fundamental skill in mathematics and various scientific fields. This methodical approach not only leads to accurate solutions but also enhances your overall understanding of algebraic principles, enabling you to approach future challenges with greater assurance and expertise. Through practice and careful application of these techniques, you can achieve mastery in algebraic problem-solving and excel in your mathematical endeavors.