Problem 71 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (2 a-3 b)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(4a^2 - 12ab + 9b^2\).

Step 1: Understand the Problem

We need to square the binomial expression \((2a - 3b)\). Squaring a binomial follows the formula \((x - y)^2 = x^2 - 2xy + y^2\).

Step 2: Apply the Squaring Formula

Identify \(x = 2a\) and \(y = 3b\) in our expression. Substitute into the formula: \((2a - 3b)^2 = (2a)^2 - 2(2a)(3b) + (3b)^2\).

Step 3: Calculate Each Term

Calculate separately: 1. \((2a)^2 = 4a^2\)2. \(-2(2a)(3b) = -12ab\)3. \((3b)^2 = 9b^2\)

Step 4: Combine the Terms

Combine all the calculated terms to get the final expression: \(4a^2 - 12ab + 9b^2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators like addition or subtraction. Think of them as sentences or phrases in math language. In our exercise, the expression is \((2a - 3b)\), which includes two terms: \(2a\) and \(-3b\). Here, \(a\) and \(b\) are variables, while \(2\) and \(-3\) are coefficients.
The beauty of algebraic expressions comes from their ability to represent general situations and to be manipulated following algebraic rules. This manipulation allows us to solve expressions or simplify them, such as transforming a complicated term into something more readable or useful.
Understanding the structure of algebraic expressions is crucial because it forms the foundation for further topics in algebra like polynomial expansion or solving equations.

Polynomial Expansion

Polynomial expansion involves taking an expression with parentheses and transforming it into a sum or a series of products without parentheses. In simpler terms, it's like opening up the brackets to see what's inside and arranging it in a readable sequence.
The binomial theorem is a cornerstone for this topic, providing a handy formula to expand expressions like \((x + y)^n\). For our exercise, we applied it to \((2a - 3b)^2\). This means identifying parts of the expression that fit the pattern and expanding them step by step.
When expanding polynomials, each term from the inner expression gets combined in every possible way with other terms. This neat process helps in revealing each component of the expression distinctly, making it easier to understand the relationships between each part.

Squaring Binomials

Squaring a binomial is a special operation in algebra where a two-term expression is multiplied by itself. In our example, the expression \((2a - 3b)\) was squared. This is a typical situation where the general formula

• \((x - y)^2 = x^2 - 2xy + y^2\)

is used. By recognizing \(x = 2a\) and \(y = 3b\), the formula directs us to compute each necessary multiplication for the expanded form.
The process includes:

• Squaring the first term: \((2a)^2\) to get \(4a^2\).
• Multiplying both terms and doubling: \(-2(2a)(3b)\) resulting in \(-12ab\).
• Squaring the second term: \((3b)^2\) resulting in \(9b^2\).

Once these calculations are done, they're combined into one seamless expression: \(4a^2 - 12ab + 9b^2\). Each step reinforces the structured approach needed in algebra to handle expressions accurately.