The **Difference of Squares** is a way of showing how an expression in the form \(x^2 - y^2\) can be written as the product of 2 **binomials**, with one as the **Sum of Square Roots** and the other as the **Difference of Square Roots**.

It can be expressed as:

\((x + y)(x - y)\)
## Factoring Difference of Squares

- Determine if there is a Greatest Common Factor (GCF) that can be factored out of both terms. Include the GCF for the remainder of the expression
- Rewrite the expression in the form \((x)^2 - (y)^2\). Do this by finding the square root of both terms
- Use the formula \((a + b)(a – b)\) to factor the rest of the expression

**Example**
Factor the binomial \(4x^2 - 25\)

We can rewrite the expression in the form \((x)^2 - (y)^2\):

\(= \sqrt{4x^2} - \sqrt{25}\)

\(= 2x - 5\)

\(= (2x)^2 - (5)^2\)
Use the formula \((x + y)(x - y)\) to factor the rest of the expression:

\(= (2x + 5)(2x - 5)\)
Therefore, we can determine that \(4x^2 - 25\) expressed as a Difference of Squares is \((2x + 5)(2x - 5)\).

Factor the following binomials:

\(121x^2 - 9y^2\)

Show Answer
We can rewrite the expression in the form \((x)^2 - (y)^2\) by taking the square root of each term then raising them to the power \(2\):

\(= (11x)^2 - (3)^2\)
Use the formula \((x + y)(x - y)\) to factor the rest of the expression:

\(= (11x + 3)(11x - 3)\)
Therefore, we can determine that \(121x^2 - 9y^2\) factored is \((11x + 3)(11x - 3)\).

\(18x^2 - 32\)

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We can factor out the GCF (in this case, 2):

\(= 2(9x^2 - 16)\)
We can rewrite the expression in the form \((x)^2 - (y)^2\) by taking the square root of each term then raising them to the power \(2\):

\(= 2((3x)^2 - (4)^2)\)
Use the formula \((x + y)(x - y)\) to factor the rest of the expression:

\(=2(3x + 4)(3x - 4)\)
Therfore, we can determine that \(18x^2 - 32\) factored is \(2(3x + 4)(3x - 4)\).

\((5c + 3)^2 - (2c + 1)^2\)

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As both terms are already squared, we can skip Step 1.

Use the formula \((x + y)(x - y)\) to factor and simplify the rest of the expression:

\(= (5c + 3) + (2c + 1))((5c + 3 - (2c + 1))\)

\(= (7c + 4)(3c + 2)\)
Therefore, we can determine that \((5c + 3)^2 - (2c + 1)^2\) factored is \((7c + 4)(3c + 2)\).

Find an algebraic expression for the area of the shaded region in factored form.

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First, we can find the area of the shaded area by finding the difference between the area of the whole shape and the area of the white area. This can be expressed as a Difference of Squares:

\(\text{Area} = (3x + 4)^2 - (x - 2)^2\)
The expression can be expanded and simplified as such:

\(\text{Area} = [(3x + 4) + (x - 2)][(3x + 4) - (x - 2)]\)

\(\text{Area} = (4x - 2)(3x + 4 - x + 2)\)

\(\text{Area} = (4x - 2)(2x + 6)\)
Take out the GCF of both binomials to further simplify the expression:

\(\text{Area} = 2(2x - 1)2(x - 3)\)

\(\text{Area} = 4(2x - 3)(x - 1)\)
Therefore, we can determine that the area of the shaded region can be expressed as \(4(2x - 3)(x - 1)\).