Problem 83 - College Algebra

Question

Perform the indicated operation or operations. $$ (3 x+4 y)^{2}-(3 x-4 y)^{2} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( (3 x+4 y)^{2}-(3 x-4 y)^{2} \) is \(24xy\).

Step 1: Identify the formula involved.

Here it is a form of a^2 - b^2 which is a difference of squares. This kind of equation can be simplified using the known algebraic formula a^2 - b^2 = (a - b)(a + b).

Step 2: Apply the formula.

Applying the algebraic formula mentioned above, we rewrite the given equation as: \((3x+4y) + (3x-4y)\) \(\cdot\) \((3x+4y) - (3x-4y)\).

Step 3: Simplify.

Simplifying the equation we get: \(2 \cdot 3x \cdot 2 \cdot 4y\).

Step 4: Simplify further.

Multiplying the constants, the equation simplifies to \(24xy\).

Key Concepts

Algebraic Simplification

When we talk about algebraic simplification, we mean reducing a complex expression to its simplest form. Think of it as cleaning up an equation to make it easier to understand and solve. In this exercise, you dealt with a special type of expression called a "difference of squares." This is where one square is subtracted from another, like \(a^2 - b^2\).

The beauty of this pattern is that it can be rewritten as a product: \((a - b)(a + b)\). This transformation makes the expression simpler and more manageable.

Here's how it works:

• Identify the squares: \( (3x+4y)^2 \) and \( (3x-4y)^2 \)
• Use the formula to rewrite the expression: \( (3x+4y) + (3x-4y) \) and \( (3x+4y) - (3x-4y) \)

This substitution is the key to simplifying the expression fully, helping us break down complex algebraic problems efficiently.

Binomial Expansion

Binomial expansion refers to expanding expressions raised to a power, like \((a+b)^n\). In this exercise, we're dealing with the expansion of binomials squared:

For example, \((3x + 4y)^2\) expands to a sum of terms. Each term involves multiplying factors from \((3x + 4y)\) by itself, according to the binomial theorem.

Consider these steps:

• Square the first term: \((3x)^2 = 9x^2\)
• Multiply and double the middle terms: \(2 imes 3x imes 4y = 24xy\)
• Square the last term: \((4y)^2 = 16y^2\)

The expansion \((3x + 4y)^2\) results in \(9x^2 + 24xy + 16y^2\). When applied to both binomials \((3x+4y)\) and \((3x-4y)\), binomial expansion helps simplify and understand the interaction of terms in the algebraic expression.

Algebraic Identities

Algebraic identities are predetermined equations true for all values of the variables involved. They are tools that help simplify and solve algebraic expressions quickly and efficiently.

The difference of squares, \(a^2 - b^2 = (a-b)(a+b)\), is a classic example. It simplifies expressions by recognizing patterns.

Why use algebraic identities?

• Speed: They make calculations faster.
• Simplicity: They reduce complex expressions to manageable chunks.
• Understanding: They reveal deep connections between different parts of algebra.

In this exercise, the identity \((a^2 - b^2)\) allowed us to rewrite \((3x+4y)^2 - (3x-4y)^2\) as \((6x)(8y)\), and ultimately simplified it to \(24xy\). By identifying and applying these identities, you can tackle various algebraic problems with confidence and ease.