Problem 59 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (a-7)^{2} $$

Step-by-Step Solution

Answer

\(a^2 - 14a + 49\)

Step 1: Expand the Expression

To expand \((a-7)^2\), we use the formula for squaring a binomial, \((x-y)^2 = x^2 - 2xy + y^2\). In this case, x = a\ and y = 7\. Applying this formula, we get:\((a-7)^2 = a^2 - 2(a)(7) + 7^2\).

Step 2: Calculate Each Term

Now, we calculate each part:1. \(a^2 = a^2\)2. \(-2(a)(7) = -14a\)3. \(7^2 = 49\).

Step 3: Assemble the Expanded Expression

Combine all parts to write the full expression:\(a^2 - 14a + 49\).

Key Concepts

Squaring a Binomial

When you encounter a binomial, such as \((x-y)^2\), and need to square it, we follow a specific formula to simplify this process. Squaring a binomial involves multiplying the binomial by itself. The general formula \((x-y)^2 = x^2 - 2xy + y^2\) makes it easy to expand these expressions without having to actually multiply them out manually.
For our exercise, we have the binomial \((a-7)^2\). Here, just replace \(x\) with \(a\) and \(y\) with \(7\).
Using the formula, you can see that:

• \(x^2\) becomes \(a^2\)
• \(-2xy\) becomes \(-2 \times a \times 7 = -14a\)
• \(y^2\) becomes \(49\)

This systematic use of the formula makes it straightforward to expand binomials in algebra. With practice, you'll find it becomes second nature.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra.
In our exercise, the expression \((a-7)^2\) involves a variable \(a\) and a constant \(7\). Once expanded, it transforms into \(a^2 - 14a + 49\), showing a clear example of how operations and variables interact within an expression.
Here's a closer look at the structure:

• **Variables**: Represent symbols like \(a\), whose values can change.
• **Constants**: Fixed values represented by numbers, such as \(7\) and \(49\).
• **Coefficients**: The numbers that appear before variables, for example, \(-14\) in \(-14a\).

Understanding the components of algebraic expressions is crucial to simplifying and manipulating them. Once you break down complex expressions into these parts, they become easier to manage and solve.

Step-by-Step Solution

Approaching binomial expansions with a step-by-step method simplifies the process and ensures accuracy. Here's a breakdown of the steps followed in solving our exercise:

**Step 1: Expand the Expression**
Apply the squaring formula \((x-y)^2 = x^2 - 2xy + y^2\). Insert the values for \(x\) and \(y\) from your binomial.
**Step 2: Calculate Each Term**
Individually compute the value for each part of the formula:

• \(a^2\) remains as \(a^2\)
• \(-2(a)(7)\) becomes \(-14a\)
• \(7^2\) simplifies to \(49\)

**Step 3: Assemble the Expression**
Combine the terms to get the full expanded form: \(a^2 - 14a + 49\). Following these steps ensures that the expansion is both systematic and comprehensible. Regular practice with these steps will enhance your understanding and confidence in handling similar algebraic tasks.