Simplifying Complex Algebraic Expressions A Step-by-Step Guide
Introduction
In this article, we will delve into the simplification of a complex algebraic expression. The expression, denoted as E, involves multiple terms, including squares, subtractions, and divisions. Our goal is to break down this expression step by step, applying algebraic identities and simplification techniques to arrive at a more concise and manageable form. This process will not only enhance our understanding of algebraic manipulations but also showcase how complex mathematical problems can be approached systematically.
The given expression is:
E = ((3x + 2y)^2 + (3x - 2y)^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
Simplifying such an expression requires careful attention to detail and a methodical approach. We will start by expanding the squared terms, combining like terms, and then dealing with the division and inverse terms. Each step will be explained in detail to ensure clarity and understanding.
Step 1: Expanding the Squared Terms
To begin, we need to expand the squared terms in the numerator. We have two terms of the form (a + b)^2 and (a - b)^2. Using the algebraic identities:
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
Applying these identities to our expression, we get:
(3x + 2y)^2 = (3x)^2 + 2(3x)(2y) + (2y)^2 = 9x^2 + 12xy + 4y^2
(3x - 2y)^2 = (3x)^2 - 2(3x)(2y) + (2y)^2 = 9x^2 - 12xy + 4y^2
Now, we substitute these expansions back into the original expression:
E = (9x^2 + 12xy + 4y^2 + 9x^2 - 12xy + 4y^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
Step 2: Combining Like Terms in the Numerator
Next, we combine the like terms in the numerator. This involves grouping terms with the same variables and powers:
9x^2 + 9x^2 - 17x^2 = (9 + 9 - 17)x^2 = x^2
12xy - 12xy = 0
4y^2 + 4y^2 - 8y^2 = (4 + 4 - 8)y^2 = 0
So, the numerator simplifies to:
x^2 + 7x + 12
Substituting this back into the expression, we have:
E = (x^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
This simplification makes the expression more manageable and easier to work with.
Step 3: Factoring the Quadratic Expressions
Now, we factor the quadratic expressions in the numerator and the denominator. Factoring helps us to identify common factors that can be canceled out, further simplifying the expression.
The numerator is a quadratic expression of the form ax^2 + bx + c, where a = 1, b = 7, and c = 12. We look for two numbers that multiply to 12 and add to 7. These numbers are 3 and 4. So, we can factor the numerator as:
x^2 + 7x + 12 = (x + 3)(x + 4)
The denominator is a difference of squares, which can be factored using the identity a^2 - b^2 = (a + b)(a - b). In our case, x^2 - 9 can be written as x^2 - 3^2. So, we factor the denominator as:
x^2 - 9 = (x + 3)(x - 3)
Substituting the factored forms back into the expression, we get:
E = ((x + 3)(x + 4)) / ((x + 3)(x - 3)) * ((x + 4) / (x - 3))^(-1)
Step 4: Dealing with the Inverse Term
The next step is to deal with the inverse term ((x + 4) / (x - 3))^(-1). Recall that a term raised to the power of -1 is the reciprocal of that term. Therefore:
((x + 4) / (x - 3))^(-1) = (x - 3) / (x + 4)
Substituting this back into the expression, we have:
E = ((x + 3)(x + 4)) / ((x + 3)(x - 3)) * (x - 3) / (x + 4)
This step is crucial as it sets the stage for further cancellations.
Step 5: Canceling Common Factors
Now, we look for common factors in the numerator and the denominator that can be canceled out. We have:
(x + 3) in the numerator and denominator
(x + 4) in the numerator and denominator
(x - 3) in the numerator and denominator
Canceling these common factors, we get:
E = ((x + 3)(x + 4)(x - 3)) / ((x + 3)(x - 3)(x + 4)) = 1
Thus, the expression simplifies to 1, provided that x ≠-3, x ≠3, and x ≠-4, as these values would make the denominator zero, rendering the expression undefined.
Conclusion
Through a series of algebraic manipulations, we have successfully simplified the given expression:
E = ((3x + 2y)^2 + (3x - 2y)^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
After expanding, combining like terms, factoring, and canceling common factors, we arrived at the simplified form:
E = 1
This exercise demonstrates the power of algebraic techniques in simplifying complex expressions. By following a step-by-step approach, we were able to break down the problem into manageable parts and arrive at a clear and concise solution. This process not only simplifies the expression but also enhances our understanding of algebraic principles and problem-solving strategies. Remember, the key to simplifying complex expressions lies in methodical application of algebraic identities and careful cancellation of terms.
Introduction to Simplifying Complex Algebraic Expressions
In mathematics, simplifying expressions is a fundamental skill that involves reducing a complex expression to its most basic form. This often makes the expression easier to understand and work with. When dealing with algebraic expressions, which can include variables, constants, and various operations, the process of simplification can be particularly challenging yet rewarding. Simplifying algebraic expressions not only helps in solving equations but also in understanding the underlying relationships between different mathematical quantities. In this comprehensive guide, we will walk through the steps required to simplify a complex expression, using the given example as a case study. The goal is to provide a clear and detailed explanation, ensuring that each step is well-understood.
The expression we aim to simplify is:
E = ((3x + 2y)^2 + (3x - 2y)^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
This expression involves several algebraic operations, including squaring binomials, adding and subtracting terms, dividing by a quadratic expression, and handling an inverse term. To simplify this, we will use various algebraic identities and techniques. Each step will be thoroughly explained to ensure clarity and comprehension. By the end of this guide, you will have a solid understanding of how to simplify complex algebraic expressions and the underlying principles involved.
Step-by-Step Breakdown of the Simplification Process
Step 1: Expanding the Squared Terms Using Algebraic Identities
The first crucial step in simplifying the expression is to expand the squared terms. This involves applying algebraic identities to remove the parentheses and make the terms easier to work with. We have two terms that need expanding: (3x + 2y)^2 and (3x - 2y)^2. These are classic examples of binomial squares, and we can use the following identities:
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
Applying these identities, we expand (3x + 2y)^2 as follows:
(3x + 2y)^2 = (3x)^2 + 2(3x)(2y) + (2y)^2
= 9x^2 + 12xy + 4y^2
Similarly, we expand (3x - 2y)^2:
(3x - 2y)^2 = (3x)^2 - 2(3x)(2y) + (2y)^2
= 9x^2 - 12xy + 4y^2
Now, we substitute these expanded forms back into the original expression. This substitution is a key part of simplifying complex equations, allowing us to work with individual terms rather than complex structures:
E = (9x^2 + 12xy + 4y^2 + 9x^2 - 12xy + 4y^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
Step 2: Combining Like Terms to Simplify the Numerator
After expanding the squared terms, the next step is to combine like terms in the numerator. This process involves identifying terms with the same variables and exponents and then adding or subtracting their coefficients. This step is critical in equation simplification because it reduces the number of terms, making the expression more manageable.
In our expression, we have several terms involving x^2, xy, and y^2. Let's group and combine these terms:
- x^2 terms: 9x^2 + 9x^2 - 17x^2 = (9 + 9 - 17)x^2 = x^2
- xy terms: 12xy - 12xy = 0
- y^2 terms: 4y^2 + 4y^2 - 8y^2 = (4 + 4 - 8)y^2 = 0
- Remaining terms: 7x + 12 (these do not have like terms to combine with)
Combining these, the numerator simplifies to:
x^2 + 7x + 12
Substituting this simplified numerator back into the expression, we have:
E = (x^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
This simplification significantly reduces the complexity of the expression, setting the stage for the next steps.
Step 3: Factoring Quadratic Expressions for Further Simplification
With the numerator now simplified to a quadratic expression, the next step is to factor it, as well as the quadratic expression in the denominator. Factoring is a crucial technique in algebraic expression simplification, as it allows us to identify common factors that can be canceled out, further reducing the expression. The numerator is x^2 + 7x + 12, which we need to factor into the form (x + a)(x + b). We look for two numbers a and b such that:
- a * b = 12
- a + b = 7
The numbers 3 and 4 satisfy these conditions, so we can factor the numerator as:
x^2 + 7x + 12 = (x + 3)(x + 4)
The denominator is x^2 - 9, which is a difference of squares. We can factor it using the identity:
a^2 - b^2 = (a + b)(a - b)
In our case, x^2 - 9 = x^2 - 3^2, so we can factor the denominator as:
x^2 - 9 = (x + 3)(x - 3)
Substituting these factored forms back into the expression, we get:
E = ((x + 3)(x + 4)) / ((x + 3)(x - 3)) * ((x + 4) / (x - 3))^(-1)
This factorization step prepares the expression for cancellation of common factors, which is the key to further simplification.
Step 4: Dealing with the Inverse Term to Facilitate Cancellation
The expression now includes an inverse term, ((x + 4) / (x - 3))^(-1). To simplify this, we need to understand that a term raised to the power of -1 is equivalent to its reciprocal. Thus,
((x + 4) / (x - 3))^(-1) = (x - 3) / (x + 4)
Substituting this back into the expression, we have:
E = ((x + 3)(x + 4)) / ((x + 3)(x - 3)) * (x - 3) / (x + 4)
This step is crucial as it sets up the expression for the final cancellation of common factors. By dealing with the inverse term, we have made it easier to identify and cancel out terms in the numerator and denominator.
Step 5: Canceling Common Factors to Achieve the Simplified Form
The final step in simplifying the expression is to cancel out any common factors that appear in both the numerator and the denominator. This process is the culmination of all the previous steps and leads us to the most simplified form of the expression. Looking at our expression:
E = ((x + 3)(x + 4)) / ((x + 3)(x - 3)) * (x - 3) / (x + 4)
We can identify the following common factors:
- (x + 3) appears in both the numerator and the denominator.
- (x + 4) appears in both the numerator and the denominator.
- (x - 3) appears in both the numerator and the denominator.
Canceling these common factors, we get:
E = ((x + 3)(x + 4)(x - 3)) / ((x + 3)(x - 3)(x + 4)) = 1
Therefore, the simplified form of the expression is 1. However, it's important to note that this simplification is valid only if x ≠-3, x ≠3, and x ≠-4, as these values would make the denominator zero at some point in the simplification process, rendering the expression undefined. The cancellation of factors in simplifying an algebraic equation is possible only if they are not zero.
Conclusion: The Power of Systematic Simplification
Through a systematic, step-by-step approach, we have successfully simplified the complex algebraic expression:
E = ((3x + 2y)^2 + (3x - 2y)^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
Our journey involved expanding squared terms, combining like terms, factoring quadratic expressions, dealing with inverse terms, and finally, canceling common factors. The result is a remarkably simple form:
E = 1
This simplification underscores the power of algebraic techniques in transforming complex expressions into manageable forms. By understanding and applying algebraic identities, factoring techniques, and the rules of reciprocals, we were able to reduce the expression to its most basic form. This process not only simplifies the expression but also enhances our understanding of algebraic principles and problem-solving strategies. Simplifying complex equations is a skill that benefits from a methodical approach and a clear understanding of basic algebraic rules.
Introduction: Unveiling the Art of Algebraic Simplification
Algebraic simplification is a cornerstone of mathematics, allowing us to transform complex expressions into simpler, more manageable forms. This skill is not only essential for solving equations but also for gaining a deeper understanding of mathematical relationships. In this guide, we will dissect a particularly intricate expression and demonstrate how to simplify it step by step. Algebraic simplification techniques can be applied to various mathematical problems.
The expression we will tackle is:
E = ((3x + 2y)^2 + (3x - 2y)^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
This expression appears daunting at first glance, with its combination of squared binomials, quadratic expressions, and inverse terms. However, by breaking it down into smaller, more manageable steps, we can systematically simplify it. Our approach will involve expanding terms, combining like terms, factoring, and canceling common factors. Each step will be thoroughly explained, ensuring that you understand not only the mechanics of simplification but also the underlying principles.
Detailed Step-by-Step Simplification Process
Step 1: Initial Expansion Using Fundamental Algebraic Identities
Our initial step involves expanding the squared terms in the numerator. This is a critical first step because it allows us to remove the parentheses and work with individual terms. We have two terms that require expansion: (3x + 2y)^2 and (3x - 2y)^2. These are classic instances where we can apply the following algebraic identities:
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
Applying these identities, we expand (3x + 2y)^2:
(3x + 2y)^2 = (3x)^2 + 2(3x)(2y) + (2y)^2
= 9x^2 + 12xy + 4y^2
Next, we expand (3x - 2y)^2:
(3x - 2y)^2 = (3x)^2 - 2(3x)(2y) + (2y)^2
= 9x^2 - 12xy + 4y^2
Now, we substitute these expanded forms back into the original expression. Substituting expanded equations is a necessary skill for simplification:
E = (9x^2 + 12xy + 4y^2 + 9x^2 - 12xy + 4y^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
Step 2: Combining Like Terms for a More Concise Numerator
After expanding the squared terms, the next logical step is to combine like terms in the numerator. This process involves identifying terms with the same variables and exponents and then adding or subtracting their coefficients. This is a key stage in algebraic simplification because it reduces the number of terms, making the expression more manageable and clearer.
In our expression, we have terms involving x^2, xy, and y^2, as well as constant terms. We group and combine these terms as follows:
- x^2 terms: 9x^2 + 9x^2 - 17x^2 = (9 + 9 - 17)x^2 = x^2
- xy terms: 12xy - 12xy = 0
- y^2 terms: 4y^2 + 4y^2 - 8y^2 = (4 + 4 - 8)y^2 = 0
- Remaining terms: 7x + 12 (these terms do not have any like terms to combine with)
Combining these simplified terms, the numerator becomes:
x^2 + 7x + 12
Substituting this simplified numerator back into the expression, we get:
E = (x^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
This simplification significantly reduces the complexity of the expression, making it easier to proceed with the next steps.
Step 3: Factoring Quadratic Expressions for Strategic Cancellation
With the numerator now simplified to a quadratic expression, the next critical step is to factor both the numerator and the denominator. Factoring is a fundamental technique in algebraic expression manipulation, as it allows us to identify common factors that can be canceled out, further simplifying the expression. The numerator is x^2 + 7x + 12, which we need to factor into the form (x + a)(x + b). We need to find two numbers a and b such that:
- a * b = 12
- a + b = 7
The numbers 3 and 4 satisfy these conditions, so we can factor the numerator as:
x^2 + 7x + 12 = (x + 3)(x + 4)
The denominator is x^2 - 9, which is a difference of squares. We can factor it using the difference of squares identity:
a^2 - b^2 = (a + b)(a - b)
In our case, x^2 - 9 = x^2 - 3^2, so we can factor the denominator as:
x^2 - 9 = (x + 3)(x - 3)
Substituting these factored forms back into the expression, we get:
E = ((x + 3)(x + 4)) / ((x + 3)(x - 3)) * ((x + 4) / (x - 3))^(-1)
This factorization step is crucial as it prepares the expression for the cancellation of common factors, which is key to further simplification.
Step 4: Addressing the Inverse Term to Facilitate Simplification
Our expression now includes an inverse term, ((x + 4) / (x - 3))^(-1). To simplify this, we need to apply the principle that a term raised to the power of -1 is equivalent to its reciprocal. Thus:
((x + 4) / (x - 3))^(-1) = (x - 3) / (x + 4)
Substituting this back into the expression, we have:
E = ((x + 3)(x + 4)) / ((x + 3)(x - 3)) * (x - 3) / (x + 4)
This step is critical in simplifying algebraic forms, as it sets the stage for the final cancellation of common factors. By dealing with the inverse term, we make it easier to identify and cancel terms in the numerator and denominator.
Step 5: Canceling Common Factors to Reveal the Simplified Solution
The final step in simplifying our expression is to cancel out any common factors that appear in both the numerator and the denominator. This process is the culmination of all the previous steps and leads us to the most simplified form of the expression. Looking at our expression:
E = ((x + 3)(x + 4)) / ((x + 3)(x - 3)) * (x - 3) / (x + 4)
We can identify the following common factors:
- (x + 3) appears in both the numerator and the denominator.
- (x + 4) appears in both the numerator and the denominator.
- (x - 3) appears in both the numerator and the denominator.
Canceling these common factors, we get:
E = ((x + 3)(x + 4)(x - 3)) / ((x + 3)(x - 3)(x + 4)) = 1
Therefore, the simplified form of the expression is 1. However, it's important to acknowledge the conditions under which this simplification is valid. The simplified expression E = 1 holds true only if x ≠-3, x ≠3, and x ≠-4. These restrictions are necessary because these values would make the denominator zero at some point during the simplification process, rendering the expression undefined. Conditions for valid equation simplification are important to consider to avoid errors.
Conclusion: Mastering Simplification for Mathematical Elegance
Through a detailed, step-by-step process, we have successfully simplified the complex algebraic expression:
E = ((3x + 2y)^2 + (3x - 2y)^2 - 17x^2 - 8y^2 + 7x + 12) / (x^2 - 9) * ((x + 4) / (x - 3))^(-1)
Our journey involved expanding squared terms, combining like terms, factoring quadratic expressions, addressing the inverse term, and ultimately, canceling common factors. The result is a remarkably simple form:
E = 1
This simplification highlights the elegance and power of algebraic techniques in transforming complex expressions into manageable forms. By carefully applying algebraic identities, factoring techniques, and the rules of reciprocals, we were able to reduce the expression to its most basic state. This process not only simplifies the expression but also deepens our understanding of algebraic principles and problem-solving strategies. Mastering the art of algebraic expression simplification involves a combination of skill, strategy, and careful execution.
