(x+a)(x-b) Formula Proof

3 min read Jun 17, 2024

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:

Expanding the product: (x + a)(x - b) = x(x - b) + a(x - b)
Applying the distributive property: = x² - bx + ax - ab
Rearranging terms: = x² + (a - b)x - ab
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.