/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Factor each expression completel... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 85

Factor each expression completely. \(\left(x^{2}+6 x+9\right)-4 y^{2}\) (Hint: Factor the trinomial in parentheses first.)

Short Answer

Expert verified
The expression factors to \((x + 3 - 2y)(x + 3 + 2y)\).

Step by step solution

01

Recognize the Structure

Notice the expression is written as a difference of two terms. Specifically, \((x^2 + 6x + 9) - 4y^2\), where the first term is a trinomial inside parentheses and the second term is a perfect square \(4y^2\).
02

Factor the Trinomial

Focus on the trinomial \(x^2 + 6x + 9\). This is a perfect square trinomial and can be factored into \((x + 3)^2\). To confirm: expand \((x + 3)^2\) to see if it equals the original trinomial. \((x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9\).
03

Recognize Difference of Squares

Now replace the trinomial with its factor in the expression: \(((x + 3)^2 - 4y^2)\). This is now a difference of squares, of the form \(a^2 - b^2\), where \(a = (x + 3)\) and \(b = 2y\).
04

Apply Difference of Squares Formula

Use the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). Substitute \(a\) and \(b\) with \((x + 3)\) and \(2y\), respectively. Thus, the expression becomes \((x + 3 - 2y)(x + 3 + 2y)\).
05

Final Expression

Confirm that the expression \((x + 3 - 2y)(x + 3 + 2y)\) is fully factored. There's no further factoring possible for these expressions.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Perfect Square Trinomial
A perfect square trinomial is a special type of quadratic expression that can be written in the form \(a^2 + 2ab + b^2\), which factors into \((a + b)^2\). Recognizing these trinomials can make factoring much simpler. For example, in the trinomial \(x^2 + 6x + 9\), we notice that:
  • The first term is \(x^2\), which is \(x\) squared.
  • The last term is \(9\), which is \(3\) squared.
  • The middle term \(6x\) is twice the product of \(x\) and \(3\).
By confirming these characteristics, we can confidently factor it as \((x + 3)^2\). It's always a good practice to expand the factors back out to check your work. Expanding \((x + 3)^2\) indeed gives you \(x^2 + 6x + 9\), verifying that this is a perfect square trinomial.
Difference of Squares
The term 'difference of squares' refers to expressions in the form \(a^2 - b^2\). This structure is unique because it can be factored using a specific formula: \(a^2 - b^2 = (a - b)(a + b)\). Recognizing when an expression can be written this way is a powerful factoring skill. Consider \(((x + 3)^2 - 4y^2)\). Here,
  • \((x + 3)^2\) represents \(a^2\) in the formula.
  • \(4y^2\) is \((2y)^2\), representing \(b^2\).
Therefore, the expression fits the difference of squares pattern. Using the formula gives us the factored form \((x + 3 - 2y)(x + 3 + 2y)\). This transformation helps simplify more complex algebraic expressions quickly.
Step-by-Step Algebra Solutions
Breaking down algebra problems into smaller, more manageable steps is essential for understanding and accuracy. Using a step-by-step approach helps in systematically simplifying and solving algebraic expressions.Here's how the original exercise was tackled:
  • Step 1: Identify the structure of the expression as a combination of different forms such as a trinomial and a perfect square.
  • Step 2: Factor the trinomial \((x^2 + 6x + 9)\) as a perfect square, \((x + 3)^2\).
  • Step 3: Recognize the format of \(a^2 - b^2\) in the new expression \(((x + 3)^2 - 4y^2)\).
  • Step 4: Apply the difference of squares formula to factor it into \((x + 3 - 2y)(x + 3 + 2y)\).
  • Step 5: Verify that the expression is fully factored and cannot be simplified further.
This process not only helps in getting the correct answer but also deepens understanding of algebraic identities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Factor each binomial completely. \(2 x^{3}+54\)

Find a positive value of \(b\) so that each trinomial is factorable. $$ m^{2}+b m-27 $$

Multiply the following. See Section 5.4 \((z-2)\left(z^{2}+2 z+4\right)\)

Factor each binomial completely. \(x^{2}-225 y^{2}\)

Find a positive value of \(c\) so that each trinomial is factorable. $$ x^{2}+6 x+c $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.