How to Factor the Difference of 2 Squares
The difference of two squares is a fundamental concept in algebra that can be found in various mathematical problems. It involves finding the factors of the expression that represents the difference between two perfect squares. In this article, we will discuss how to factor the difference of two squares and provide some examples to illustrate the process.
The general form of a difference of two squares is:
\[ a^2 – b^2 \]
To factor this expression, we can use the following formula:
\[ a^2 – b^2 = (a + b)(a – b) \]
This formula can be derived from the algebraic identity:
\[ (a + b)(a – b) = a^2 – b^2 \]
Now, let’s go through the steps to factor the difference of two squares:
1. Identify the two perfect squares in the expression.
2. Apply the formula \( (a + b)(a – b) \) by replacing \( a \) with the larger perfect square and \( b \) with the smaller perfect square.
3. Multiply the resulting binomials to obtain the factored form.
Let’s look at some examples:
Example 1:
Factor the expression \( 16 – 9 \).
Step 1: Identify the two perfect squares: \( 16 \) and \( 9 \).
Step 2: Apply the formula: \( (4 + 3)(4 – 3) \).
Step 3: Multiply the binomials: \( 7 \times 1 = 7 \).
So, the factored form of \( 16 – 9 \) is \( 7 \).
Example 2:
Factor the expression \( 25 – 64 \).
Step 1: Identify the two perfect squares: \( 25 \) and \( 64 \).
Step 2: Apply the formula: \( (5 + 8)(5 – 8) \).
Step 3: Multiply the binomials: \( -3 \times 3 = -9 \).
So, the factored form of \( 25 – 64 \) is \( -9 \).
In conclusion, factoring the difference of two squares is a straightforward process that involves identifying the two perfect squares and applying the formula \( (a + b)(a – b) \). By following these steps, you can easily factor any expression that represents the difference of two squares.
