Solve the Equation: 1/(x-2)² + 1/(x-2) = 1 Step-by-Step

Solve the following equation:

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

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

• Step 1: Identify the expressions $\frac{1}{(x-2)^2}$ and $\frac{1}{x-2}$.
• Step 2: Combine the fractions by using a common denominator.
• Step 3: Multiply through by the common denominator and simplify.
• Step 4: Rearrange the resulting equation to form a quadratic equation.
• Step 5: Solve the quadratic equation using the quadratic formula.

Carrying out these steps:

Step 2: The common denominator is $(x-2)^2$. Rewrite the equation as:
$\frac{1}{(x-2)^2} + \frac{x-2}{(x-2)^2} = 1$.

Step 3: Combine the fractions:
$\frac{1 + (x-2)}{(x-2)^2} = 1$.

Step 3: Simplifying gives:
$\frac{x-1}{(x-2)^2} = 1$.

Step 3: Cross-multiply to eliminate the fraction:
$x - 1 = (x-2)^2$.

Step 4: Expand the right-hand side:
$x - 1 = x^2 - 4x + 4$.

Step 4: Rearrange to form a quadratic equation:
$x^2 - 5x + 5 = 0$.

Step 5: Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a = 1$, $b = -5$, $c = 5$:
$x = \frac{5 \pm \sqrt{25 - 20}}{2}$.

Step 5: Simplify:
$x = \frac{5 \pm \sqrt{5}}{2}$.

This results in two potential solutions for $x$:
$x = \frac{1}{2}[5+\sqrt{5}]$ and $x = \frac{1}{2}[5-\sqrt{5}]$.

Therefore, the solution to the problem is $x = \frac{1}{2}[5 \pm \sqrt{5}]$, which matches the correct answer choice.