In Exercises 39–48, factor the difference of two squares. 36x²−49y²

Verified step by step guidance

1. Recognize that the given expression, $36x^2\; -\; 49y^2$, is a difference of two squares. The general formula for factoring the difference of two squares is $a^2\; -\; b^2\; =\; (a\; -\; b)(a\; +\; b)$.
2. Identify the square terms in the expression. Here, $36x^2$ is the square of $6x$, and $49y^2$ is the square of $7y$.
3. Rewrite the expression in terms of its squared components: $(6x)^2\; -\; (7y)^2$.
4. Apply the difference of squares formula: $(6x)^2\; -\; (7y)^2\; =\; (6x\; -\; 7y)(6x\; +\; 7y)$.
5. Write the factored form of the expression as $(6x\; -\; 7y)(6x\; +\; 7y)$.

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic expression that takes the form a^2 - b^2, which can be factored into (a - b)(a + b). This concept is essential for simplifying expressions and solving equations, as it allows for the breaking down of complex quadratic forms into simpler linear factors.

Factoring

Factoring is the process of breaking down an expression into its constituent parts or factors that, when multiplied together, yield the original expression. In the context of polynomials, factoring is crucial for simplifying expressions, solving equations, and understanding the roots of the polynomial.

Quadratic Expressions

Quadratic expressions are polynomial expressions of the form ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. Understanding quadratic expressions is vital for recognizing patterns such as the difference of squares, which can lead to efficient factoring and solving techniques in algebra.