Lesson: Expanding an Expression to a Difference of Two Squares | Nagwa - Free Printable

The image you've uploaded presents a mathematical concept related to expanding expressions using the FOIL method and specifically highlights the difference of two squares identity.

Let's break it down and explain the solution step by step.

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🔷 What is FOIL?

FOIL is a mnemonic used to multiply two binomials:

- First: Multiply the first terms in each binomial
- Outside: Multiply the outer terms
- Inside: Multiply the inner terms
- Last: Multiply the last terms in each binomial

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🔷 The Expression Given:

$$
(x + y)(x - y)
$$

We are asked to expand this expression using FOIL and show that it results in a difference of two squares.

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🔷 Step-by-Step Expansion Using FOIL:

Let’s apply FOIL to $(x + y)(x - y)$:

1. First: $x \cdot x = x^2$
2. Outside: $x \cdot (-y) = -xy$
3. Inside: $y \cdot x = xy$
4. Last: $y \cdot (-y) = -y^2$

Now, add all these together:

$$
x^2 - xy + xy - y^2
$$

Notice that $-xy + xy = 0$, so they cancel out.

$$
x^2 - y^2
$$

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✔ Final Result:

$$
(x + y)(x - y) = x^2 - y^2
$$

This is known as the difference of two squares identity.

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🔷 Why Is This Important?

This identity is very useful in algebra for:
- Factoring expressions like $x^2 - 9$ → $(x + 3)(x - 3)$
- Simplifying complex expressions
- Solving equations

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🟩 Summary:

✔ The expression $(x + y)(x - y)$ expands to $x^2 - y^2$ using the FOIL method.
✔ This is called the difference of two squares.
✔ It's a fundamental algebraic identity.

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🧠 Example:

Let’s try with numbers:
$(5 + 3)(5 - 3) = 8 \cdot 2 = 16$
And $5^2 - 3^2 = 25 - 9 = 16$ → same result!

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✔ So, the image correctly illustrates the identity:
$$
\boxed{(x + y)(x - y) = x^2 - y^2}
$$

This is a key algebraic tool!