Problem 96 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (3 a-5 b)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \((3a - 5b)^2\). Answer: \(9a^2 - 30ab + 25b^2\).

Step 1: Apply the expansion formula

Recall the formula \((a - b)^2 = a^2 - 2ab + b^2\). Apply this formula to the given expression \((3a - 5b)^2\) to expand it.

Step 2: Substitute the values

Replace a and b with the expressions provided. In our case, \(a = 3a\) and \(b = 5b\). The expansion formula then becomes $$ (3a - 5b)^2 = (3a)^2 - 2(3a)(5b) + (5b)^2 $$

Step 3: Simplify the expression

Simplify the expression by multiplying the terms and combining like terms, if any: $$ (3a)^2 - 2(3a)(5b) + (5b)^2 = 9a^2 - 30ab + 25b^2 $$ The simplified algebraic expression is \(9a^2 - 30ab + 25b^2\).

Key Concepts

Expansion Formula

When simplifying algebraic expressions, understanding the expansion formula is critical. This formula is a useful tool for multiplying and simplifying expressions, especially when dealing with binomials. A binomial is simply a polynomial with two terms.

One common form of the expansion formula is for squaring a binomial, which follows the pattern \( (a \pm b)^2 = a^2 \pm 2ab + b^2 \). When you see an expression like \( (3a - 5b)^2 \), you can apply this pattern. Here's a breakdown:

• The first term in the binomial is squared: \( 3a \times 3a = 9a^2 \).
• The two terms are multiplied together, doubled, and reflect the sign between them: \( 2 \times 3a \times 5b = 30ab \), and with a minus sign in the original expression, it becomes \( -30ab \).
• The second term is squared: \( 5b \times 5b = 25b^2 \).

Using the expansion formula properly ensures that every term is accounted for and the simplification process is accurate.

Binomial Squares

Squaring binomials is a specific application of the expansion formula. Recognizing binomial squares can help you simplify algebraic expressions more efficiently. Let's revisit our example \( (3a - 5b)^2 \). This is a binomial square because it involves raising a binomial to the power of two.

Understanding this concept allows you to visualize each component of the binomial as something that needs to be squared or doubled and then combined.

What Happens During Squaring?

When we square a binomial:

• The first term is squared, producing a term with an exponent of two.
• The product of the two terms is doubled.
• The second term is squared.

It's vital to maintain the correct signage throughout the process. In our case, the negative sign between \(3a\) and \(5b\) means we subtract when combining \(9a^2\) and \(25b^2\), keeping the \(30ab\) with a negative sign.

Algebraic Operations

Knowing how to perform algebraic operations is essential when working with any algebraic expression. These operations include addition, subtraction, multiplication, and the distribution of terms.

Once the expansion formula is applied to expressions like \( (3a - 5b)^2 \), we need to perform the multiplication and combine like terms. Here's what happens:

• Multiply powers of the same base by adding their exponents.
• Multiply coefficients (numerical values) separately from the variables.
• Look for like terms—terms that have the same variables raised to the same power—and combine them by adding or subtracting.

However, our example doesn't have like terms to combine after expansion, which is a common scenario with binomial squares<.>
Through these operations, we obtain our simplified expression \(9a^2 - 30ab + 25b^2\). Each step in the process follows the order of operations to ensure accuracy.