Problem 71
Question
Simplify. $$\frac{(x-1)^{2}}{(x+1)^{2}}-1$$
Step-by-Step Solution
Verified Answer
The simplified form of the expression is \(\frac{-2x}{(x+1)^{2}}\).
1Step 1: Identifying Mathematical Properties
The given expression can be seen as \( \frac{a^2}{b^2} - 1 \), where \( a = (x-1) \) and \( b = (x+1) \). This suggests the possibility of using the formula for the difference of two squares: \( \frac{a^2}{b^2} - 1 = (\frac{a}{b} -1)(\frac{a}{b} + 1) \).
2Step 2: Applying the Difference of Squares Formula
Inserting the values of \( a = x-1 \) and \( b = x+1 \) into the formula, the expression becomes \( (\frac{x-1}{x+1} -1)(\frac{x-1}{x+1} + 1) \).
3Step 3: Simplifying the Two Binomials
\(\frac{x-1}{x+1} -1 = \frac{x-1-x}{x+1} = \frac{-1}{x+1} \) and \( \frac{x-1}{x+1} +1 = \frac{x-1+x}{x+1} = \frac{2x}{x+1} \). Therefore, the expression simplifies to \( \frac{-1}{x+1} * \frac{2x}{x+1} \).
4Step 4: Solving the simplified expression
Now we multiply the two fractions, \( \frac{-1 * 2x}{(x+1)*(x+1)} = \frac{-2x}{(x+1)^{2}} \).
Key Concepts
Simplifying ExpressionsDifference of SquaresFractions in Algebra
Simplifying Expressions
Simplifying expressions is a fundamental concept in prealgebra. When simplifying expressions, you aim to make them easier to work with by reducing them to their simplest form. This typically involves using various algebraic operations to combine like terms and eliminate any unnecessary parts of the expression.
In the exercise given, we utilized the process of simplification on an algebraic fraction. Recognizing that the expression was in a certain setup, \( \\frac{a^2}{b^2} - 1\), guided us toward using different algebraic identities, like the difference of squares, which further simplified our original expression. Always begin by looking for parts of the expression that are common, or can be combined or cancelled out, which in this case was achieved by applying the mathematical identity.Understanding the format and identifying opportunities to simplify expressions not only makes the expressions less complex but also prepares you for more challenging problems in algebra.
In the exercise given, we utilized the process of simplification on an algebraic fraction. Recognizing that the expression was in a certain setup, \( \\frac{a^2}{b^2} - 1\), guided us toward using different algebraic identities, like the difference of squares, which further simplified our original expression. Always begin by looking for parts of the expression that are common, or can be combined or cancelled out, which in this case was achieved by applying the mathematical identity.Understanding the format and identifying opportunities to simplify expressions not only makes the expressions less complex but also prepares you for more challenging problems in algebra.
Difference of Squares
The difference of squares is an essential algebraic identity useful in simplifying expressions. It is expressed as \( a^2 - b^2 = (a - b)(a + b)\). Understanding this identity allows you to transform certain complex expressions into simpler, more manageable forms.
In our exercise, this identity was incredibly valuable. By recognizing the original expression's structure as something similar to \( \frac{a^2}{b^2} - 1\), we could express it as \( (\frac{a}{b} - 1)(\frac{a}{b} + 1)\). This makes our problem significantly easier as it transforms a seemingly complex fraction problem into the multiplication of two simpler binomials.
The difference of squares works because of the predictable result when multiplying the two conjugate pairs \( (a - b)\) and \( (a + b)\). These always yield \( a^2 - b^2\), no matter the values of \( a \) or \( b\). This identity simplifies our task by utilizing inherent mathematical symmetry.
In our exercise, this identity was incredibly valuable. By recognizing the original expression's structure as something similar to \( \frac{a^2}{b^2} - 1\), we could express it as \( (\frac{a}{b} - 1)(\frac{a}{b} + 1)\). This makes our problem significantly easier as it transforms a seemingly complex fraction problem into the multiplication of two simpler binomials.
The difference of squares works because of the predictable result when multiplying the two conjugate pairs \( (a - b)\) and \( (a + b)\). These always yield \( a^2 - b^2\), no matter the values of \( a \) or \( b\). This identity simplifies our task by utilizing inherent mathematical symmetry.
Fractions in Algebra
Fractions in algebra involve variables in either the numerator, the denominator, or both. They follow the rules of fractions with numbers, with additional attention to the variables. Simplifying algebraic fractions can involve factoring, cancelling terms, and, importantly, improving mathematical operations like multiplication or division.
In the original exercise, we worked with algebraic fractions by breaking down the original expression into components where simple, known rules of fractions applied. By expressing the fraction \( \frac{x-1}{x+1} - 1\) in terms of common denominators, we simplified it to separate terms that were easier to manage. Simplifying the components individually allowed us to express them in simpler terms such as \( \frac{-1}{x+1} \) and \( \frac{2x}{x+1} \).
When you understand fractions and their manipulations, algebra becomes a more predictable and less daunting subject. Learning to handle fractions with ease enhances your overall algebra skills, as fractions are ubiquitous in mathematics and science.
In the original exercise, we worked with algebraic fractions by breaking down the original expression into components where simple, known rules of fractions applied. By expressing the fraction \( \frac{x-1}{x+1} - 1\) in terms of common denominators, we simplified it to separate terms that were easier to manage. Simplifying the components individually allowed us to express them in simpler terms such as \( \frac{-1}{x+1} \) and \( \frac{2x}{x+1} \).
When you understand fractions and their manipulations, algebra becomes a more predictable and less daunting subject. Learning to handle fractions with ease enhances your overall algebra skills, as fractions are ubiquitous in mathematics and science.
Other exercises in this chapter
Problem 70
Simplify. $$\frac{2 x}{x^{2}-x-6}-\frac{3}{x+2}$$
View solution Problem 70
Divide. $$\frac{4 a x-8 a}{c^{2}} \div \frac{2 y-x y}{c^{3}}$$
View solution Problem 71
Divide. $$\frac{3 x^{2} y-9 x y}{a^{2} b} \div \frac{3 x^{2}-x^{3}}{a b^{2}}$$
View solution Problem 72
Simplify. $$1-\frac{(y-2)^{2}}{(y+2)^{2}}$$
View solution