Solving the Equation \( x + \frac{1}{x} = 2 \) to Evaluate Algebraic Expressions

Let's solve this mathematical problem step-by-step, starting with the given equation: \( x + \frac{1}{x} = 2 \). Our goal is to find the numerical value of the expression \( x^{99} + \frac{1}{x^{99}} - 2 \).

Step 1: Determine the Value of x from the Given Equation

We are given the equation:

\( x + \frac{1}{x} = 2 \)

To find the value of x, we can clear the fraction by multiplying every term in the equation by x. We should note that if x were 0, the expression \(\frac{1}{x}\) would be undefined, so x cannot be 0. Let's proceed assuming x is not zero:

\( x \cdot \left( x + \frac{1}{x} \right) = 2 \cdot x \)

Distribute x on the left side:

\( x \cdot x + x \cdot \frac{1}{x} = 2x \)

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

Now, rearrange the terms to form a standard quadratic equation by moving the 2x term to the left side:

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

This equation is a perfect square trinomial. It can be factored into the square of a binomial:

\( (x - 1)^2 = 0 \)

To solve for x, take the square root of both sides:

\( \sqrt{(x - 1)^2} = \sqrt{0} \)

\( x - 1 = 0 \)

Finally, solve for x:

\( x = 1 \)

Since \( x = 1 \), which is not zero, our initial assumption that x is not zero was valid.

Step 2: Substitute \( x = 1 \) into the Expression \( x^{99} + \frac{1}{x^{99}} - 2 \)

Now that we have found the value of x, we can substitute \( x = 1 \) into the given expression:

The expression is \( x^{99} + \frac{1}{x^{99}} - 2 \).

Substitute \( x = 1 \):

\( (1)^{99} + \frac{1}{(1)^{99}} - 2 \)

Step 3: Evaluate the Resulting Expression

We know that any positive integer power of 1 is always 1. Therefore, \( 1^{99} = 1 \).

Substitute this back into the expression:

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

Simplify the fraction:

\( 1 + 1 - 2 \)

Perform the addition and subtraction:

\( 2 - 2 \)

\( 0 \)

Conclusion

When \( x + \frac{1}{x} = 2 \), the value of the expression \( x^{99} + \frac{1}{x^{99}} - 2 \) is 0.

Revision Table: Key Derivation from \( x + \frac{1}{x} = 2 \)

This table summarizes the crucial step of solving for x:

Additional Information: Powers of 1 and Related Algebraic Identities

Understanding the properties of the number 1 is very helpful in solving problems like this:

• Any positive integer power of 1 is 1: \( 1^n = 1 \) for any positive integer n.
• The reciprocal of 1 is 1: \( \frac{1}{1} = 1 \).

Also, recognizing the algebraic identity \( (a - b)^2 = a^2 - 2ab + b^2 \) was key in factoring the quadratic equation \( x^2 - 2x + 1 = 0 \) as \( (x - 1)^2 = 0 \). Problems involving \( x + \frac{1}{x} \) often relate to powers of x or standard algebraic identities.