Question

Factor the expression completely.

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

Short answer

The solution ${\left(1+\frac{1}{x}\right)}^2-{\left(1-\frac{1}{x}\right)}^2=\frac{4}{x}$

Step-by-step solution

Step 1. Apply the “difference of square of sum and square of difference” formula.

If A and B are any real numbers or algebraic expressions, then

${\left(A+B\right)}^2-{\left(A-B\right)}^2=4AB$

Step 2. Analyze the given expression.

Given ${\left(1+\frac{1}{x}\right)}^2-{\left(1-\frac{1}{x}\right)}^2$

Here, $A=1,B=\frac{1}{x}$

Step 3. Solve the expression.

Substitute the values in difference of square of sum and square of difference formula:

$\begin{array}{l}{\left(1+\frac{1}{x}\right)}^2-{\left(1-\frac{1}{x}\right)}^2=4AB\\ \Rightarrow {\left(1+\frac{1}{x}\right)}^2-{\left(1-\frac{1}{x}\right)}^2=4\cdot 1\cdot \frac{1}{x}\\ \Rightarrow {\left(1+\frac{1}{x}\right)}^2-{\left(1-\frac{1}{x}\right)}^2=\frac{4}{x}\end{array}$