Problem 100 - College Algebra

Question

Perform the indicated operations. $$(x-y)^{2}-(x+y)^{2}$$

Step-by-Step Solution

Answer

- 4xy

Step 1: Expand the Squares

The first thing to do is expand \((x-y)^2\) and \((x+y)^2\). Using the formula \(a^2 - b^2\), which expands to \((a - b)(a + b)\). Here, a = x-y and b = x+y. \((x-y)^{2}\) expands to \(x^{2} - 2xy + y^{2}\) and \((x+y)^{2}\) expands to \(x^{2} + 2xy + y^{2}\).

Step 2: Apply Subtraction

Next, subtract \((x+y)^{2}\) from \((x-y)^{2}\). So, from \(x^{2} - 2xy + y^{2}\) subtract \(x^{2} + 2xy + y^{2}\) to get \(x^{2} - 2xy + y^{2} - x^{2} - 2xy - y^{2}\).

Step 3: Simplification

Now, simplify the equation by cancelling out similar terms to get the final answer. The terms \(- x^{2}\) and \(x^{2}\) cancel each other out to become 0. Similarly, \(y^{2}\) and \(- y^{2}\) also cancel each other out to become 0. Hence, the result leaves only \(- 2xy - 2xy\).

Step 4: Final Solution

\(- 2xy - 2xy\), simplifies to \(- 4xy\), this is your final answer.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical statements that involve numbers, variables, and operations. Variables are symbols like \(x\) and \(y\) that can represent different numbers. In an expression like \((x-y)^2 - (x+y)^2\), parentheses indicate that the enclosed terms should be treated as a single unit during operations. This is crucial when performing expansions or other operations.

Understanding the roles of each component in an algebraic expression helps in performing accurate operations.

• Variables represent unknown values or a set of numbers.
• Constants are numbers that always maintain the same value.
• Operators like addition, subtraction, multiplication and division describe relationships between numbers or variables.

Practicing with algebraic expressions allows deeper comprehension of how numbers and operations behave collectively. It is a foundational skill that underpins more complex mathematical concepts.

Polynomial Operations

Polynomial operations involve performing algebraic manipulations like addition, subtraction, multiplication, and division on polynomials. Polynomials are expressions with multiple terms, where each term consists of variables raised to a power and multiplied by a coefficient.

In the exercise, you expand the polynomials \((x-y)^2\) and \((x+y)^2\) by using formulae known for these expansions:

• \((a-b)^2 = a^2 - 2ab + b^2\)
• \((a+b)^2 = a^2 + 2ab + b^2\)

It's essential to analyze and understand each term's contribution when expanding polynomials. After expanding, subtraction of one polynomial from another is performed. Combining like terms involves adjusting terms that have the same variables raised to the same power. This simplifies the expression and moves one step closer to finding the solution. Becoming skilled in polynomial operations opens the door to solving more complex equations and expressions.

Quadratic Expressions

Quadratic expressions are specific polynomials of degree 2, often written in the form \(ax^2 + bx + c\). These expressions describe parabolas when graphed and are foundational in both algebra and calculus studies. In our exercise, the expressions \((x-y)^2\) and \((x+y)^2\) began as quadratic before expanding and simplifying.

The expansion steps described are part of transforming a quadratic into a simpler, more manageable form.

• The quadratic expression \((x-y)^2\) yields \(x^2 - 2xy + y^2\).
• The expression \((x+y)^2\) yields \(x^2 + 2xy + y^2\).

When you subtract these, the like terms \(x^2\) and \(y^2\) cancel out, leaving us with linear terms as a result. Understanding how to manipulate quadratic expressions is crucial, as they appear widely across various mathematical problems and real-life applications. Mastery over quadratic expressions not only aids in simple exercises but also in more complex problem-solving scenarios.