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Chapter 4: Problem 73

Factor. \((2 x+5)^{2}-x^{2}\)

Short Answer

Expert verified
The factored form of \((2x+5)^2-x^2\) is \((3x+5)(x+5)\).

Step by step solution

01

Identify the Difference of Squares

Recognize that the expression \((2x+5)^2 - x^2\) is a difference of squares. The formula for the difference of squares is \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a = 2x + 5\) and \(b = x\).
02

Apply the Difference of Squares Formula

Use the identified values for \(a\) and \(b\) in the difference of squares formula. Substitute \(a = 2x + 5\) and \(b = x\) to get:\[(2x + 5 + x)(2x + 5 - x)\].
03

Simplify Each Binomial

Simplify the expressions within each parenthesis:1. For \((2x + 5 + x)\), combine like terms: \(2x + x + 5 = 3x + 5\).2. For \((2x + 5 - x)\), combine like terms: \(2x - x + 5 = x + 5\).
04

Write the Final Factored Form

Use the simplified binomials from Step 3 to express the factored form of the original expression: \((3x + 5)(x + 5)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expressions
Binomial expressions are a foundational concept in algebra. They are algebraic expressions containing exactly two terms, often joined by a plus or minus sign. For example, in the exercise where we dealt with \((2x + 5)^2 - x^2\), the components \(2x + 5\) and \(x\) are each individual terms forming part of bigger binomial expressions when they involve operations like squaring or subtracting.
Binomials are critical in pattern recognition, especially when factoring complex equations. Learning to recognize and manipulate these expressions allows students to simplify and solve various algebraic problems effectively.
Algebraic Identities
Algebraic identities are crucial tools in algebra that provide reliable ways to transform expressions. They are statements equating one algebraic expression to another, consistently and universally true.
A prime example of an algebraic identity is the difference of squares, which states:
  • \(a^2 - b^2 = (a + b)(a - b)\)
This identity helps in quickly factoring expressions into predictable products of binomials.
In our problem, identifying the expression \((2x + 5)^2 - x^2\) as a difference of squares enabled applying this identity to factor it with ease. Recognizing these patterns is immensely helpful in working efficiently through algebraic problems.
Simplifying Expressions
Simplifying expressions is a key process in algebra to make equations more manageable. It involves reducing an expression to its simplest form while ensuring it represents the same value.
In the exercise, after applying the algebraic identity for the difference of squares, we simplified the resulting expression \( (2x + 5 + x)(2x + 5 - x) \).
  • First, we combined like terms in each binomial: \(2x + x + 5 = 3x + 5\)
  • And \(2x - x + 5 = x + 5\)
This simplification is crucial as it leads to a cleaner, clearer expression \((3x + 5)(x + 5)\), providing an accurate and concise solution. By practicing simplification, students develop skills to tackle more complex algebraic equations with confidence.

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