DIFFERENCE BETWEEN SQUARES $$ \begin{array}{l}x^{2}-1 \ \frac{a^{4}}{16}-\frac{1}{25} b^{2} \ x^{2 a}-1\end{array} \begin{array}{l}x^{2}\left(a^{2}-9 ight)+y^{2}\left(-a^{2}+9 ight)\end{array} $$

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DIFFERENCE BETWEEN SQUARES $$ \begin{array}{l}x^{2}-1 \ \frac{a^{4}}{16}-\frac{1}{25} b^{2} \ x^{2 a}-1\end{array} \begin{array}{l}x^{2}\left(a^{2}-9ight)+y^{2}\left(-a^{2}+9ight)\end{array} $$

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The **difference of squares** is a fundamental algebraic identity that allows us to factor expressions of the form $$ a^2 - b^2 $$ into $$ (a + b)(a - b) $$. This technique is widely used to simplify and solve various algebraic expressions. Let's apply this method to each of the given expressions.

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### 1. $$ x^{2} - 1 $$

**Step 1:** Recognize the expression as a difference of squares.

$$
x^{2} - 1 = x^{2} - 1^{2}
$$

**Step 2:** Apply the difference of squares formula.

$$
x^{2} - 1 = (x + 1)(x - 1)
$$

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### 2. $$ \frac{a^{4}}{16} - \frac{1}{25} b^{2} $$

**Step 1:** Rewrite each term as a square.

$$
\frac{a^{4}}{16} = \left( \frac{a^{2}}{4} \right)^{2}, \quad \frac{1}{25} b^{2} = \left( \frac{b}{5} \right)^{2}
$$

**Step 2:** Express the original expression as a difference of squares.

$$
\frac{a^{4}}{16} - \frac{1}{25} b^{2} = \left( \frac{a^{2}}{4} \right)^{2} - \left( \frac{b}{5} \right)^{2}
$$

**Step 3:** Apply the difference of squares formula.

$$
\frac{a^{4}}{16} - \frac{1}{25} b^{2} = \left( \frac{a^{2}}{4} + \frac{b}{5} \right) \left( \frac{a^{2}}{4} - \frac{b}{5} \right)
$$

**Optional:** To simplify further, you can find a common denominator for each factor.

$$
\left( \frac{5a^{2} + 4b}{20} \right)\left( \frac{5a^{2} - 4b}{20} \right) = \frac{(5a^{2} + 4b)(5a^{2} - 4b)}{400}
$$

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### 3. $$ x^{2a} - 1 $$

**Step 1:** Recognize the expression as a difference of squares by identifying $$ x^{a} $$ as the base.

$$
x^{2a} - 1 = \left( x^{a} \right)^{2} - 1^{2}
$$

**Step 2:** Apply the difference of squares formula.

$$
x^{2a} - 1 = \left( x^{a} + 1 \right) \left( x^{a} - 1 \right)
$$

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### 4. $$ x^{2}(a^{2} - 9) + y^{2}(-a^{2} + 9) $$

**Step 1:** Simplify the expression by factoring out common terms.

$$
x^{2}(a^{2} - 9) + y^{2}(-a^{2} + 9) = x^{2}(a^{2} - 9) - y^{2}(a^{2} - 9) = (a^{2} - 9)(x^{2} - y^{2})
$$

**Step 2:** Notice that both $$ a^{2} - 9 $$ and $$ x^{2} - y^{2} $$ are differences of squares.

$$
a^{2} - 9 = a^{2} - 3^{2} = (a + 3)(a - 3)
$$

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

**Step 3:** Combine the factorizations.

$$
x^{2}(a^{2} - 9) + y^{2}(-a^{2} + 9) = (a + 3)(a - 3)(x + y)(x - y)
$$

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By systematically applying the difference of squares formula, we've successfully factored each of the given expressions into simpler, multiplicative forms.
The difference of squares can be used to factor expressions of the form $$ a^2 - b^2 $$ into $$ (a + b)(a - b) $$. Applying this to the given expressions: 1. $$ x^{2} - 1 = (x + 1)(x - 1) $$ 2. $$ \frac{a^{4}}{16} - \frac{1}{25} b^{2} = \left( \frac{a^{2}}{4} + \frac{b}{5} \right) \left( \frac{a^{2}}{4} - \frac{b}{5} \right) $$ 3. $$ x^{2a} - 1 = (x^{a} + 1)(x^{a} - 1) $$ 4. $$ x^{2}(a^{2} - 9) + y^{2}(-a^{2} + 9) = (a + 3)(a - 3)(x + y)(x - y) $$ These factorizations simplify the original expressions by breaking them down into their constituent parts.

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