Introduction
When dealing with algebraic expressions, factoring is a crucial skill that allows us to simplify and solve complex equations. In this article, we will dive into the factored form of the expression N^2 – 25, understand the process of factoring, and explore different methods to factorize the given expression.
Understanding Factoring
Factoring is the process of breaking down an algebraic expression into its simplest form by identifying its factors. Factors are numbers or expressions that when multiplied together result in the original expression. Factoring is a fundamental technique in algebra and plays a significant role in solving equations and simplifying expressions.
The Expression N^2 – 25
The expression N^2 – 25 can be written in the form of a difference of squares. In algebra, a difference of squares is an expression that can be factored into the product of two conjugate binomials. The expression N^2 – 25 fits this pattern as it can be expressed as (N + 5)(N – 5).
Factoring N^2 – 25
To factor the expression N^2 – 25, we can follow the steps for factoring a difference of squares:
- Identify the expression: The given expression is N^2 – 25.
- Recognize the pattern: N^2 – 25 can be expressed as N^2 – 5^2, which is the form of a difference of squares.
- Apply the formula: The formula to factor a difference of squares is (a^2 – b^2) = (a + b)(a – b). In this case, a = N and b = 5.
- Factor the expression: Substituting the values of a and b into the formula, we get (N + 5)(N – 5) as the factored form of N^2 – 25.
Therefore, the factored form of N^2 – 25 is (N + 5)(N – 5).
Verification of the Factored Form
To verify that the factored form (N + 5)(N – 5) is correct, we can expand the expression and simplify it back to the original form:
- Multiplying the binomials: (N + 5)(N – 5) = N^2 – 5N + 5N – 25
- Simplifying the expression: Combining like terms, we get N^2 – 25
The factored form (N + 5)(N – 5) correctly expands to N^2 – 25, confirming the accuracy of the factoring process.
Applications of Factoring
Factoring is a fundamental skill in algebra that has various applications in mathematics, physics, engineering, and other fields. Some of the common applications of factoring include:
- Solving equations: Factoring is used to solve quadratic equations and express them in simpler forms.
- Simplifying expressions: Factoring helps simplify complex algebraic expressions and make calculations easier.
- Graphing functions: Factoring can help identify x-intercepts and factors of a function for graphing purposes.
- Finding roots: Factoring is essential in finding the roots or solutions of polynomial equations.
Conclusion
In conclusion, the factored form of the expression N^2 – 25 is (N + 5)(N – 5). By understanding the process of factoring, recognizing patterns in algebraic expressions, and applying the appropriate formulas, we can simplify complex equations and solve problems efficiently. Factoring is a fundamental skill in mathematics with diverse applications in various fields, making it an essential tool for problem-solving and mathematical analysis.
By mastering the concept of factoring and practicing different factoring techniques, we can enhance our algebraic skills and tackle challenging mathematical problems with confidence. The factored form of N^2 – 25 serves as a prime example of how factoring can simplify expressions and lead to accurate solutions in algebra.
