Exploring the Relationship Between a2 - b2 and Given Fractions: A Comprehensive Guide
In mathematics, particularly in algebra, understanding the relationship between expressions such as a2 - b2 and given fractions can be quite revealing. This article will delve into the concept of a2 - b2, explore its relationship with different fractions, and provide step-by-step instructions on how to solve problems revolving around such expressions.
Introduction
The expression a2 - b2 is a fundamental concept in algebra, often referred to as the difference of squares. This expression can be simplified and manipulated in various ways, which will be explored in the sections to follow. We will also demonstrate how such expressions can be solved within the context of a given fraction, such as 13/12.
Simplification and Solution of the Given Expression
Consider the following equation:
3a 4b 13
This equation can be derived from the given condition (frac{a}{4} frac{b}{3} frac{13}{12}).
To solve for a2 - b2, we first need to express b in terms of a. By manipulating the equation, we get:
b frac{13 - 3a}{4}).
Substituting this into the expression for a2 - b2, we obtain:
.
Now, let's simplify this expression:
.
.
The above simplification leads us to:
.
Finally, by substituting a 3 and b 1, we find:
.
Which simplifies to:
10.
Algebraic Identities and Their Applications
The expression a2 - b2 can also be represented using algebraic identities. Here are two common identities:
Identity1:
a2b2 a2 - b2 - 2ab
Identity2:
a2b2 (a - b)2 - 2ab
Using these identities, we can derive the value of a2 - b2. For example, using Identity1:
.
Given a 3 and b 1, we get:
.
Which simplifies to:
10.
Conclusion
Through the analysis and exploration of the given problem, we have successfully determined that the value of a2 - b2 is 10. This result was achieved by simplifying the expression, substituting values, and applying algebraic identities. Understanding these methods can greatly assist in solving similar problems involving the difference of squares and given fractions.
Related Keywords
a squared b squared
algebraic identities
fraction simplification