Understanding the (x + a)(x - b) Formula
The formula (x + a)(x - b) = x² - (b - a)x - ab is a fundamental algebraic identity used to expand and simplify expressions. It's crucial in various mathematical contexts, including solving equations, factoring expressions, and working with quadratic equations.
Proof:
We can prove this formula using the distributive property of multiplication. Here's the breakdown:
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Expanding the product: (x + a)(x - b) = x(x - b) + a(x - b)
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Applying the distributive property: = x² - bx + ax - ab
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Rearranging terms: = x² + (a - b)x - ab
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Final step: = x² - (b - a)x - ab
Therefore, the formula (x + a)(x - b) = x² - (b - a)x - ab is proven.
Key Points:
- Sign convention: Notice that the coefficient of the x term is (b - a), not (a - b), which is crucial to maintain the correct sign.
- Product of constants: The constant term in the expanded expression is the product of the constants, -ab.
- Application: This formula is widely used for factoring quadratic expressions, simplifying algebraic expressions, and even deriving other important identities in algebra.
Example:
Let's apply the formula to expand (x + 3)(x - 2):
Using the formula, we have:
(x + 3)(x - 2) = x² - (2 - 3)x - (3 * 2) = x² + x - 6
Therefore, the expanded form of (x + 3)(x - 2) is x² + x - 6.
Conclusion:
The formula (x + a)(x - b) = x² - (b - a)x - ab is a powerful tool in algebra, offering a shortcut for expanding and simplifying expressions. Its understanding and application are essential for mastering algebraic concepts and solving various mathematical problems.