Problem 62 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((2 y+5)(2 y-5)\)

Step-by-Step Solution

Answer

The simplified expression is \(4y^2 - 25\).

Step 1: Identify the Product Type

Recognize that the expression \((2y + 5)(2y - 5)\) fits the formula of the difference of squares, which is given by the identity \((a + b)(a - b) = a^2 - b^2\). In this case, \(a = 2y\) and \(b = 5\).

Step 2: Apply the Special Product Formula

Using the difference of squares formula \((a + b)(a - b) = a^2 - b^2\), we substitute \(a = 2y\) and \(b = 5\) into the formula to get: \((2y)^2 - 5^2\).

Step 3: Calculate Each Square

Calculate \((2y)^2 = 4y^2\) and \(5^2 = 25\) by squaring each term individually. This results in: \(4y^2 - 25\).

Step 4: Write the Simplified Expression

Combine the calculated results to express the simplified version of the product: \(4y^2 - 25\).

Key Concepts

Special Product Formula

The special product formula for finding the product of two binomials that involve a sum and a difference, such as \((a + b)(a - b)\), is known as the difference of squares formula. This particular formula simplifies the multiplication process because it identifies a pattern inherent in the binomials.This formula can be exceptionally useful, as it states that:\[(a + b)(a - b) = a^2 - b^2\]

• Recognizing the Pattern: Notice that the binomials are similar except for the sign between the terms (one is a sum, the other a difference).
• Application: This identity allows us to quickly find the product without having to use long multiplication.
• Examples of Use: Commonly, expressions like \((x + y)(x - y)\) or \((2a + 3b)(2a - 3b)\) fit this pattern perfectly, demonstrating the power of this formula.

Understanding this special product formula saves time and simplifies calculations when dealing with algebraic expressions. In our example, \((2y + 5)(2y - 5)\) becomes more manageable by applying this formula. This derivation showcases the elegance of algebraic shortcuts.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a key skill in algebra. It involves reducing expressions to their simplest form without changing their value, which makes them easier to work with and solve.In the context of the exercise, once you apply the difference of squares formula, you are left with \((2y)^2 - 5^2\). This expression can further be simplified by calculating each square.

• Calculate Individual Elements: Compute \((2y)^2\) which yields \(4y^2\), and \(5^2\) which results in \(25\). These calculations transform the expression into \(4y^2 - 25\).
• Combine and Simplify: The expression \(4y^2 - 25\) is now fully simplified given that there are no like terms to combine further.

Simplifying helps in reducing complexity and often paves the way for solving equations or comparing expressions more efficiently.

Algebraic Multiplication

Algebraic multiplication involves multiplying numbers, variables, or combinations of both, known as terms. Understanding this concept is essential for dealing with more complex expressions and solving algebraic problems.In our example \((2y + 5)(2y - 5)\), algebraic multiplication occurs between two binomials:

• Expanding the Binomials: Typically, we would use the distributive property to expand, but the special product formula streamlines this process by directly reaching the result.
• Result Interpretation: Multiplying these binomials using the formula immediately gives us \(4y^2 - 25\). This result shows how powerful and efficient algebraic multiplication can be with the proper techniques.

Mastering algebraic multiplication, especially through the use of special products and strategies, enables quicker and more accurate problem-solving in algebra. This application demonstrates the strength of practice and understanding fundamentals to tackle multiply-layered expressions.