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

4 min read Jun 17, 2024

The (x+a)(x+b) Formula and Its Proof

The formula (x+a)(x+b) = x² + (a+b)x + ab is a fundamental algebraic expression that simplifies the multiplication of two binomials. It's a useful tool for expanding and manipulating algebraic expressions. Let's explore this formula and its proof in detail.

Understanding the Formula

The formula states that the product of two binomials, (x+a) and (x+b), is equal to the sum of three terms:

x²: The product of the first terms of each binomial (x * x).
(a+b)x: The sum of the products of the outer and inner terms (ax + bx).
ab: The product of the last terms of each binomial (a * b).

Proof of the Formula

We can prove this formula using the distributive property of multiplication:

Distribute the first binomial: (x+a)(x+b) = x(x+b) + a(x+b)
Distribute again: x(x+b) + a(x+b) = x² + xb + ax + ab
Rearrange the terms: x² + xb + ax + ab = x² + (a+b)x + ab

Therefore, we have proven that (x+a)(x+b) = x² + (a+b)x + ab

Example

Let's illustrate this formula with an example:

(x+2)(x+3)

Using the formula:

x²: x * x = x²
(a+b)x: (2+3)x = 5x
ab: 2 * 3 = 6

Therefore:

(x+2)(x+3) = x² + 5x + 6

Applications

The (x+a)(x+b) formula is widely used in various mathematical and scientific contexts, including:

Factoring quadratic expressions: This formula helps to factor quadratic expressions into their binomial forms.
Solving quadratic equations: The formula can be used to express quadratic equations in a simplified form for easier solving.
Simplifying algebraic expressions: The formula is useful for expanding and simplifying complex algebraic expressions involving binomials.
Developing other mathematical formulas: This formula serves as a foundation for deriving other important mathematical formulas and theorems.

Conclusion

The (x+a)(x+b) formula is a crucial algebraic tool that provides a shortcut for multiplying two binomials. Its simplicity and wide applicability make it a valuable asset for understanding and solving various mathematical problems. Remember this formula and practice applying it to different situations to enhance your algebraic skills.