factorise the following using the difference of two squares rule: x^2-1

To factorize the expression \(x^2 - 1\) using the difference of two squares rule, follow these steps:

### Step 1: Recognize the form
The difference of two squares rule states that:
\[
a^2 - b^2 = (a - b)(a + b)
\]
In the expression \(x^2 - 1\), notice that:
- \(x^2\) is a perfect square (\(x^2 = (x)^2\))
- \(1\) is also a perfect square (\(1 = (1)^2\))

### Step 2: Apply the difference of two squares rule
Using the rule, substitute \(a = x\) and \(b = 1\) into the formula:
\[
x^2 - 1 = (x - 1)(x + 1)
\]

### Final Answer:
\[
x^2 - 1 = (x - 1)(x + 1)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you identify expressions that can be factored using the difference of two squares?
2. Can the difference of two squares rule be used for any two squares, regardless of the variables?
3. How would you factorize \(4x^2 - 9\) using the difference of two squares rule?
4. What happens when there is an additional term in the expression, such as \(x^2 - 1 + 2x\)?
5. How would you apply the difference of two squares rule to an expression like \(x^4 - 16\)?

### Tip:
The difference of two squares can only be used when both terms are perfect squares and the sign between them is a minus. If you have a plus sign, other factoring methods must be used.

Learn how to factorize the expression x^2 - 1 using the difference of two squares rule with a detailed step-by-step solution. This problem covers key algebraic concepts including factorization, quadratic equations, and the difference of two squares formula. Suitable for students in Grades 7-9.

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