Problem 22 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ 49-a^{2} $$

Step-by-Step Solution

Answer

The factored form of the expression \(49 - a^{2}\) is \((7 - a)(7 + a)\).

Step 1: Identify a and b

In our expression, \(49 - a^{2}\), we recognize that 49 is a perfect square. Thus, \(a\) is 7 and \(b\) is \(a\).

Step 2: Apply difference of squares formula

We apply the difference of squares formula and substitute our values of \(a\) and \(b\) into it. The formula is \(a^{2} - b^{2} = (a - b)(a + b)\), which therefore yields \((7 - a)(7 + a)\).

Key Concepts

Difference of Squares

The Difference of Squares is a special algebraic expression that can be factored into two binomials. It takes the form of \(a^2 - b^2\). The magic behind this expression lies in its simplicity to be broken down into

• \((a - b)(a + b)\), which is very straightforward to remember.

If you see two perfect squares, separated by a minus sign, you have likely encountered a difference of squares.
As applied in our problem, \(49 - a^2\), we recognize it as a difference of squares. This is because both 49 and \(a^2\) are perfect squares.
49 is a perfect square because it is \(7^2\), and \(a^2\) is simply \(a^2\). Hence, the expression \(49 - a^2\) becomes \((7 - a)(7 + a)\) when factored.
This method is efficient and saves time, especially in longer algebraic manipulations.

Perfect Squares

A Perfect Square, in algebra, refers to the product of an integer with itself. Common example: \(x^2\) is a perfect square because it is simply \(x\times x\).
In our case, the number 49 is a perfect square, since 49 equals \(7^2\).
Recognizing perfect squares assists in simplifying and solving algebraic expressions easily.
Consider the expression: 64, \(x^2\), 100. All are perfect squares because

• 64 is \(8^2\),
• \(x^2\) is \(x\times x\),
• 100 is \(10^2\).

Knowing this helps you identify potential uses of the difference of squares formula and other factoring techniques.

Algebraic Expression

An Algebraic Expression is a mathematical phrase that includes numbers, variables, and operators. It’s a toolkit for expressing various quantities.
The expression \(49 - a^2\) is a simple algebraic expression involving a subtraction of squares.
Understanding how to manipulate these expressions is essential for solving equations and simplifying problems.
Algebraic expressions can look like:

• \(3x + 4\)
• \(x^2 - 5x + 6\)
• \(a^2 - b^2\)

By identifying the key components, like perfect squares or recognizing all sorts of patterns, you can factor the expressions easier and work through problems faster.