Problem 55 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((2 x+3 y)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(4x^2 + 12xy + 9y^2\).

Step 1: Identify the Special Product Formula

The expression is a binomial squared: \((2x + 3y)^2\). It fits the form of \((a + b)^2\), where \(a = 2x\) and \(b = 3y\). The special product formula for \((a + b)^2\) is \(a^2 + 2ab + b^2\).

Step 2: Apply the Special Product Formula

Using the formula \(a^2 + 2ab + b^2\), substitute \(a = 2x\) and \(b = 3y\) into the formula. This gives us: \((2x)^2 + 2(2x)(3y) + (3y)^2\).

Step 3: Calculate Each Term

Compute each part of the expanded expression: - \((2x)^2 = 4x^2\), - \(2(2x)(3y) = 12xy\), - \((3y)^2 = 9y^2\).

Step 4: Write the Simplified Result

Combine the calculated terms to write the simplified expression: \(4x^2 + 12xy + 9y^2\).

Key Concepts

Binomial Squares

When you come across an algebraic expression where you see something like \((a + b)^2\), you are dealing with a binomial square. Binomials are simply expressions that contain two terms, or in other words, 'bi' for two. Here, the two terms are multiplied by themselves or squared. Let's break down its meaning.

In the expression \((2x + 3y)^2\), the term is squared, which means we multiply it by itself. To simplify this multiplying process, we apply the special product formula instead of expanding it the long way. The special product formula for a binomial square is \( (a + b)^2 = a^2 + 2ab + b^2 \).

Using this formula, we can expand and simplify binomial squares very quickly without doing the multiplication part step-by-step each time.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols.These expressions are fundamental in algebra and can vary from simple forms like \(3x + 2\) to more complex ones such as \((2x + 3y)^2\).

In an algebraic expression, each symbol has a role:

• **Numbers** are constants or coefficients in expressions (like the 3 in \(3x\)),
• **Variables** represent unknown values (like x and y in \(2x+3y\)),
• **Operations** like addition, subtraction, multiplication guide how to combine our constants and variables (`+`, `-`, `×`).

Understanding the composition of an algebraic expression helps to maneuver more easily when applying formulas and performing operations, like in our binomial squares.

Simplifying Expressions

Simplifying expressions means rewriting them in a more straightforward or cleaner form without changing their value or meaning. It often involves reducing the number of terms, combining like terms, or removing parentheses by performing operations.

To simplify the expression \( (2x + 3y)^2 \), we first identify it as a binomial square and apply the special product formula. The formula gives us the terms:

• \( 4x^2 \) from \((2x)^2\),
• \( 12xy \) from \(2(2x)(3y)\),
• \( 9y^2 \) from \((3y)^2\).

These calculations give the simplified format: \(4x^2 + 12xy + 9y^2\).

The essence of simplifying expressions is to express them as cleanly as possible with the fewest steps. That makes it easier to work further with them in algebraic equations and problem-solving.