Problem 44 - Elementary Algebra

Question

Factor completely. $$ (x-5) 2-(x-2) 2 $$

Step-by-Step Solution

Answer

The expression factors to \((2x - 7)(-3)\).

Step 1: Recognize the Formula

The expression \((x-5)^2-(x-2)^2\) matches the form of a difference of squares: \(a^2 - b^2\). Recall that this can be factored as \((a+b)(a-b)\).

Step 2: Identify the Components

Assign \(a = (x-5)\) and \(b = (x-2)\). This means that the expression \((x-5)^2-(x-2)^2\) is now ready to be expressed using the formula for difference of squares.

Step 3: Apply the Difference of Squares Formula

Apply the formula \((a+b)(a-b)\) using the identified components. We have \((x-5)+(x-2)\) and \((x-5)-(x-2)\).

Step 4: Simplify Each Factor

Simplify each part of the expression: -\((x-5)+(x-2) = x - 5 + x - 2 = 2x - 7\)-\((x-5)-(x-2) = x - 5 - x + 2 = -3\).

Step 5: Write the Final Factored Form

Combine the simplified expressions to write the completely factored expression: \[(2x - 7)(-3)\]

Step 6: Final Simplification (Optional)

The expression \((2x - 7)(-3)\) can be further simplified by distributing the -3: \[2x(-3) - 7(-3) = -6x + 21\]. However, the original problem asked for factoring, so \[(2x - 7)(-3)\] is already the completely factored form.

Key Concepts

Difference of Squares

The concept of a difference of squares is an essential idea in algebra. It refers to expressions that take the form \(a^2 - b^2\). This expression is called "difference of squares" because it is the result of subtracting one square from another. This particular structure allows for a straightforward factorization method.

When you recognize an expression as a difference of squares, you can apply a specific factoring formula. This formula states that \(a^2 - b^2 = (a + b)(a - b)\). This means you take the square root of each square term to find \(a\) and \(b\), then use them to construct the factored terms. The process results in two binomials whose product gives the original expression back.

Mastering the identification of the difference of squares can greatly simplify solving algebraic problems, as it transforms complex expressions into much simpler factors.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the foundation of algebra. In our original problem, the expression \((x-5)^2-(x-2)^2\) is an example of such an algebraic expression.

These expressions can appear complex, but breaking them down into manageable parts helps in simplification and manipulation. Each part of the expression holds particular significance:

• The term \((x-5)\) represents a variable part that is squared.
• Similarly, \((x-2)\) also represents a squared variable term.

Recognizing such parts enables effective application of mathematical techniques like factoring, which is crucial in solving algebraic equations.

Factoring Formulas

Factoring formulas are specific techniques used to rewrite expressions as a product of simpler expressions. These are essential tools in algebra because they simplify and solve equations effectively.

The difference of squares is one of the basic types of factoring formulas, and it goes as \(a^2 - b^2 = (a + b)(a - b)\). By applying this formula, we can express a complex expression in a simpler multiplicative form.

For example, in the expression \((x-5)^2 - (x-2)^2\), identifying \(a\) as \((x-5)\) and \(b\) as \((x-2)\), the difference of squares formula can be directly applied. This simplifies the expression into two binomials \((a + b)(a - b)\), which can be further simplified by calculating each term. Understanding and using these formulas allows for greater versatility in handling various algebraic problems.

Simplification Process

The simplification process is an essential step in algebra that converts expressions into their simplest forms. It involves reducing complexity, whether by factoring or other methods, to make expressions more manageable. In the original problem, the expression \((x-5)^2-(x-2)^2\) was reduced using the difference of squares method.

After factoring, you need to simplify further by evaluating each component. For instance, when applying the difference of squares:

• First, calculate \((x-5) + (x-2)\) to get \(2x - 7\).
• Next, calculate \((x-5) - (x-2)\) to obtain \(-3\).

These calculations are part of the simplification process that reduces expressions to their simplest factored forms. Understanding this process builds a solid foundation for tackling more complex algebraic tasks effectively.