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Simplifying the Expression: (a - 2)(a + 4) - (a + 1)^2
This article will guide you through simplifying the algebraic expression (a - 2)(a + 4) - (a + 1)^2. We will use the distributive property and the FOIL method to expand the terms and then combine like terms to reach a simplified expression.
Expanding the Terms
First, we need to expand the terms using the distributive property:
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(a - 2)(a + 4): We multiply each term in the first set of parentheses by each term in the second set:
- a * a = a²
- a * 4 = 4a
- -2 * a = -2a
- -2 * 4 = -8
- (a - 2)(a + 4) = a² + 4a - 2a - 8
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(a + 1)²: This term represents the square of a binomial, so we can use the FOIL method (First, Outer, Inner, Last):
- First: a * a = a²
- Outer: a * 1 = a
- Inner: 1 * a = a
- Last: 1 * 1 = 1
- (a + 1)² = a² + a + a + 1
Combining Like Terms
Now we can substitute the expanded expressions back into the original equation and simplify:
(a - 2)(a + 4) - (a + 1)² = (a² + 4a - 2a - 8) - (a² + a + a + 1)
Combine like terms within each set of parentheses:
(a² + 2a - 8) - (a² + 2a + 1)
Finally, subtract the terms within the second set of parentheses from the first set:
a² + 2a - 8 - a² - 2a - 1
= -9
Conclusion
Therefore, the simplified form of the expression (a - 2)(a + 4) - (a + 1)² is -9. This means the expression will always equal -9 regardless of the value of 'a'.