Solving Algebraic Expressions: Finding the Value of (m - 2)² + 1/(m - 2)²

We are given the equation: \[m + \frac{1}{m - 2} = 4\] We need to find the value of the expression: \[(m - 2)^2 + \frac{1}{(m - 2)^2}\]

Step-by-Step Solution

Manipulating the Given Equation

Let's rearrange the given equation to make it easier to work with. Notice that the expression we need to find involves \((m - 2)\). Let's try to get \((m - 2)\) in the given equation.

Subtract 2 from both sides of the equation \(m + \frac{1}{m - 2} = 4\): \[m + \frac{1}{m - 2} - 2 = 4 - 2\] Rearranging the terms on the left side gives: \[(m - 2) + \frac{1}{m - 2} = 2\]

Introducing a Substitution

To simplify the notation, let's substitute \(x = m - 2\). The equation now becomes: \[x + \frac{1}{x} = 2\] The expression we need to find the value of is \((m - 2)^2 + \frac{1}{(m - 2)^2}\), which in terms of \(x\) is: \[x^2 + \frac{1}{x^2}\]

Using an Algebraic Identity

We know the algebraic identity for the square of a sum: \((a + b)^2 = a^2 + 2ab + b^2\).

Let's apply this identity to \((x + \frac{1}{x})^2\), where \(a = x\) and \(b = \frac{1}{x}\): \[\left(x + \frac{1}{x}\right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2\] \[\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}\]

We want to find the value of \(x^2 + \frac{1}{x^2}\). We can rearrange the identity to isolate this term: \[x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2\]

Substituting and Calculating

We found from the given equation that \(x + \frac{1}{x} = 2\).

Now, substitute this value into the expression for \(x^2 + \frac{1}{x^2}\): \[x^2 + \frac{1}{x^2} = (2)^2 - 2\] \[x^2 + \frac{1}{x^2} = 4 - 2\] \[x^2 + \frac{1}{x^2} = 2\]

Since \(x = m - 2\), this means: \[(m - 2)^2 + \frac{1}{(m - 2)^2} = 2\]

Conclusion

The value of the expression \((m - 2)^2 + \frac{1}{(m - 2)^2}\) is 2.

Let's quickly check the options provided:

Our calculated value is 2, which matches Option 4.

Revision Table: Key Algebraic Concepts

Additional Information: Solving \(x + \frac{1}{x} = 2\)

It is interesting to note that the equation \(x + \frac{1}{x} = 2\) has a very specific solution.

We can solve this equation for \(x\): Multiply both sides by \(x\) (assuming \(x \neq 0\)): \[x \cdot \left(x + \frac{1}{x}\right) = 2 \cdot x\] \[x^2 + 1 = 2x\] Rearrange into a quadratic equation: \[x^2 - 2x + 1 = 0\] This is a perfect square trinomial, which can be factored as: \[(x - 1)^2 = 0\] Taking the square root of both sides: \[x - 1 = 0\] \[x = 1\]

So, the only value of \(x\) that satisfies \(x + \frac{1}{x} = 2\) is \(x = 1\). Since \(x = m - 2\), this means \(m - 2 = 1\), which gives \(m = 3\).

We could also find the value of \((m - 2)^2 + \frac{1}{(m - 2)^2}\) by directly substituting \(m - 2 = 1\): \[(1)^2 + \frac{1}{(1)^2} = 1 + \frac{1}{1} = 1 + 1 = 2\] This confirms our result obtained using the identity method. This detailed look provides a deeper understanding of the values of \(m\) and \(m-2\) involved.