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Chapter 5: Problem 78

Factor. $$16 m^{2}-24 m-n^{2}+9$$

Short Answer

Expert verified
The factorized form of the given expression \[16m^{2}-24m-n^{2}+9\] is \[(4m-3+n)(4m-3-n)\]

Step by step solution

01

Rearrange the terms

First identify the like terms and rearrange them in the order of ascending or descending powers. In this case, the given expression is \[16m^{2}-24m-n^{2}+9\]. We rearrange it as \[16m^{2}-24m+9-n^{2}\]
02

Identify the perfect squares

Next observe that \(16m^{2}-24m+9\) forms a perfect square. It can be transformed into \((4m-3)^{2}\) using the formula \(a^{2}-2ab+b^{2}=(a-b)^{2}\) if we identify \(4m\) as 'a' and '3' as 'b'.
03

Representation as a difference of squares

Further, observe that the modified expression becomes \((4m-3)^{2}-n^{2}\). This now looks like a difference of squares which can be simplified using the formula \(a^{2}-b^{2}=(a+b)(a-b)\). Here, we can consider \((4m-3)\) as 'a' and 'n' as 'b'.
04

Applying the difference of squares rule

By applying the difference of squares rule, we finally obtain \((4m-3+n)(4m-3-n)\)

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Key Concepts

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

Difference of Squares
The difference of squares is a fundamental concept in algebra that involves subtracting one square number from another. It's expressed in the form \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\). This method is useful for simplifying complex algebraic expressions.
  • Identify two perfect squares within your expression.
  • Write each term as a square number.
  • Apply the formula for the difference of squares.
Consider the expression \((4m-3)^2 - n^2\). Here, \((4m - 3)^2\) and \(n^2\) are perfect squares. This expression can be directly rewritten as \((4m-3+n)(4m-3-n)\), illustrating the beauty of this factorization technique.
Perfect Square Trinomial
A perfect square trinomial is an expression that can be written as the square of a binomial. The general form is \(a^2 - 2ab + b^2 = (a - b)^2\). Recognizing these trinomials is key to simplifying or factoring expressions in algebra.
  • Look for expressions with three terms, usually in squares.
  • Verify if it follows the pattern \(a^2 - 2ab + b^2\).
  • Factor into \((a - b)^2\) or \((a + b)^2\).
In the expression \(16m^2 - 24m + 9\), we can identify it as a perfect square trinomial because:- \(a = 4m\)- \(b = 3\)This trinomial can be simplified to \((4m - 3)^2\), achieving a simpler representation.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to make calculations easier and find solutions more efficiently. It's a core skill in algebra that enables problem-solving across various mathematical contexts.
  • Rearrange terms to recognize patterns such as perfect squares.
  • Look for opportunities to apply known formulas like the difference of squares or perfect square trinomials.
  • Simplify expressions to facilitate further calculations.
Starting with the expression \(16m^2 - 24m - n^2 + 9\), algebraic manipulation allowed us to transform it into \((4m-3)^2 - n^2\). This involved identifying and reordering terms to recognize the patterns of a perfect square trinomial and a difference of squares, highlighting the importance of strategic rearrangement in algebra.

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Most popular questions from this chapter

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