Problem 46 - Elementary Algebra

Question

Factor completely. $$ 25 x 2-36 $$

Step-by-Step Solution

Answer

The expression factors to \((5x - 6)(5x + 6)\).

Step 1: Identify the Expression

We start with the expression \( 25x^2 - 36 \). We will factor this expression completely.

Step 2: Recognize the Form

Notice that \( 25x^2 - 36 \) is a difference of squares. Recall that a difference of squares can be factored using the formula \( a^2 - b^2 = (a-b)(a+b) \).

Step 3: Write as a Difference of Squares

Express \( 25x^2 \) as \((5x)^2\) and \( 36 \) as \(6^2\). So, the given expression matches the form \((5x)^2 - 6^2\).

Step 4: Apply the Difference of Squares Formula

Use the formula \( a^2 - b^2 = (a-b)(a+b) \) where \( a = 5x \) and \( b = 6 \). Substitute these into the formula to get \((5x - 6)(5x + 6)\).

Step 5: Verify the Factorization

Expand \((5x - 6)(5x + 6)\) to confirm it simplifies back to \( 25x^2 - 36 \). The multiplication gives: \( (5x)(5x) + (5x)(6) - (6)(5x) - (6)(6) = 25x^2 - 36 \), confirming the factorization is correct.

Key Concepts

Difference of Squares

The concept of a difference of squares is a handy tool in algebra for simplifying expressions. It involves recognizing an expression that can be written in the form \( a^2 - b^2 \). This specific form can be easily factored using the formula:

• \( a^2 - b^2 = (a - b)(a + b) \)

When approaching an expression like \( 25x^2 - 36 \), it's important to notice that it is structured as a difference of squares.
Here, \( 25x^2 \) can be rewritten as \((5x)^2\) and \( 36 \) as \( 6^2 \). By recognizing this, you can swiftly factor the original expression.
Understanding difference of squares is a foundational algebra skill that simplifies solving and factoring complex algebraic expressions.

Complete Factorization

Complete factorization refers to breaking down an algebraic expression into the product of its simplest factors.
For the expression \( 25x^2 - 36 \), we begin by identifying it as a difference of squares, which immediately gives us a straightforward path to factor:

• Using \((5x)^2 - 6^2\), apply the difference of squares formula to obtain \((5x - 6)(5x + 6)\).

The expression is then considered completely factored when no further factorization is possible.
Complete factorization helps in simplifying simplifying terms and solving algebraic equations by reducing them to their simplest forms.
Once you get the hang of these patterns, your problem-solving speed and confidence in algebra will improve significantly.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of algebra.
In exercises like the factorization of \( 25x^2 - 36 \), understanding algebraic expressions is crucial.
Here's what algebraic expressions typically include:

• **Variables**: Symbols like \( x \) that represent numbers.
• **Coefficients**: Numbers that multiply variables, like \( 25 \) in \( 25x^2 \).
• **Constants**: Numbers on their own, like \( 36 \).

Breaking down expressions into simpler parts - like identifying squares in \( 25x^2 - 36 \) - makes it easier to work with them.
Understanding how to manipulate algebraic expressions using techniques like factorization is key to solving many types of algebra problems effectively.