How to factor the difference of two squares?

Hello there! 👋 It's fantastic you're diving into factoring, and the 'difference of two squares' is a super important pattern to master. Don't worry, it's actually one of the easiest and most common factoring techniques once you see how it works! Let's break it down in a friendly, clear way.

What is the Difference of Two Squares? 🤔

As the name suggests, the 'difference of two squares' is an algebraic expression where you have one perfect square term subtracted from another perfect square term. It's always in the form of something squared minus something else squared.

The general form is: $a^2 - b^2$

Here, '$a$' and '$b$' can be any numbers, variables, or even entire expressions themselves! The key is that both terms are perfect squares, and there's a minus sign between them.

The Magical Formula ✨

The coolest part about the difference of two squares is that it always factors into a very specific, predictable pattern:

$a^2 - b^2 = (a - b)(a + b)$

Notice how it's always one factor with a minus sign and another with a plus sign, using the 'square roots' of the original terms.

Why Does This Work? A Quick Peek! 👀

Let's quickly multiply $(a - b)(a + b)$ using the FOIL method (First, Outer, Inner, Last) to see why it equals $a^2 - b^2$:

• First: $a \cdot a = a^2$
• Outer: $a \cdot b = ab$
• Inner: $-b \cdot a = -ab$
• Last: $-b \cdot b = -b^2$

When you put it all together: $a^2 + ab - ab - b^2$. See how the middle terms ($+ab$ and $-ab$) cancel each other out? That leaves you with just $a^2 - b^2$! Pretty neat, right? 😎

Let's See Some Examples! 🚀

Example 1: Simple Case

Factor: $x^2 - 25$

1. Identify $a^2$ and $b^2$: Here, $a^2 = x^2$ (so $a = x$) and $b^2 = 25$ (so $b = 5$).
2. Apply the formula: $(a - b)(a + b)$
3. Substitute $a$ and $b$: $(x - 5)(x + 5)$

So, $x^2 - 25 = (x - 5)(x + 5)$. Easy peasy!

Example 2: With Coefficients and Higher Powers

Factor: $4y^2 - 81z^4$

1. Identify $a^2$ and $b^2$: This one looks a bit more complex, but the process is the same!
   • $a^2 = 4y^2 \Rightarrow a = \sqrt{4y^2} = 2y$
   • $b^2 = 81z^4 \Rightarrow b = \sqrt{81z^4} = 9z^2$
2. Apply the formula: $(a - b)(a + b)$
3. Substitute $a$ and $b$: $(2y - 9z^2)(2y + 9z^2)$

Therefore, $4y^2 - 81z^4 = (2y - 9z^2)(2y + 9z^2)$.

Example 3: Don't Forget the GCF!

Factor: $3x^3 - 12x$

Sometimes you need to factor out a Greatest Common Factor (GCF) first! Always check for that before applying other factoring methods.

1. Find the GCF: The GCF of $3x^3$ and $12x$ is $3x$.
2. Factor out the GCF: $3x(x^2 - 4)$
3. Now, notice that the expression inside the parentheses, $(x^2 - 4)$, is a difference of two squares! ($a=x$, $b=2$)
4. Factor the difference of two squares: $3x(x - 2)(x + 2)$

So, $3x^3 - 12x = 3x(x - 2)(x + 2)$.

Key Takeaways for Success:

• Look for Subtraction: It MUST be a difference (subtraction), not a sum ($a^2 + b^2$ cannot be factored over real numbers!).
• Check for Perfect Squares: Both terms must be perfect squares.
• Always Check for GCF First: This simplifies the problem significantly.

You got this! Practice a few of these, and you'll be factoring the difference of two squares like a pro in no time. Let me know if any other questions pop up! 😊