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Factoring the Difference of Squares: (x-10)(x+10)
The expression (x-10)(x+10) is a prime example of a common algebraic pattern known as the difference of squares. This pattern occurs when we have two binomials, one with a positive term and the other with a negative term, where the terms are identical except for the sign.
Recognizing the Difference of Squares
To identify the difference of squares, look for these key characteristics:
- Two binomials: The expression should be the product of two sets of parentheses.
- Identical terms: The terms inside the parentheses should be the same, but with opposite signs.
In our example, (x-10)(x+10):
- We have two binomials: (x-10) and (x+10).
- The terms inside the parentheses are identical: 'x' and '10', with opposite signs in each binomial.
Applying the Pattern
The difference of squares pattern simplifies to:
(a - b)(a + b) = a² - b²
where 'a' and 'b' represent the terms in the binomials.
Let's apply this pattern to our expression:
- Identify 'a' and 'b': In our case, 'a' = x and 'b' = 10.
- Substitute into the formula: (x - 10)(x + 10) = x² - 10²
- Simplify: x² - 10² = x² - 100
The Result
Therefore, the simplified form of (x-10)(x+10) is x² - 100.
This pattern is extremely useful in simplifying expressions and solving equations. It highlights the power of recognizing and utilizing common algebraic patterns.