Problem 68 Factor. $$x^{2}+6 x y+9 y^{2}$... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 68

Factor. $$x^{2}+6 x y+9 y^{2}$$

Short Answer

Expert verified

The factored form of the quadratic expression $x^{2}+6xy+9y^{2}$ is $(x+3y)^{2}$.

Step by step solution

Identify coefficients

First, identify the coefficients of the equation. In the quadratic expression $x^{2}+6xy+9y^{2}$, the coefficient of $x^{2}$ is 1, the coefficient of the $xy$ term is 6, and the coefficient of $y^{2}$ is 9.

Check if the equation is a perfect square trinomial

Recognize that the equation is in the form of $a^{2} + 2ab + b^{2}$, which suggests that it's a perfect square trinomial. We can see that $(\sqrt{1})^{2} +2* \sqrt{1}*\sqrt{9} + (\sqrt{9})^{2}$ gives us the original expression. Thus, the equation is indeed a perfect square trinomial where $a = x$ and $b = 3y$.

Apply the formula

Finally, apply the formula for perfect square trinomial which is $(a+b)^{2}$. here, $a = x$ and $b = 3y$, so substituting these values in the formula, we get $(x+3y)^{2}$

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 quadratic expression that can be expressed as the square of a binomial. This means it has the form $a^2 + 2ab + b^2$. When an expression fits this pattern, you can factor it into $(a + b)^2$.

In the example $x^2 + 6xy + 9y^2$, the expression matches the perfect square trinomial form where:

$a = x$
$b = 3y$

This means that the trinomial is $(x + 3y)^2$. Recognizing this pattern makes factoring faster and simpler!

Quadratic Expression

Quadratic expressions are polynomial expressions of degree two鈥攜ou'll often see them in the form $ax^2 + bx + c$. The highest power of the variable is 2, which makes it a quadratic. These expressions are foundational in algebra and appear in a variety of mathematical problems.

Our specific expression $x^2 + 6xy + 9y^2$ is a quadratic, even though there are two variables, $x$ and $y$. The key feature is the $x^2$ term, identifying it as quadratic in $x$. Quadratics can be graphed as parabolas, and knowing how to factor them helps identify roots and critical points of these graphs.

Coefficients

Coefficients are the numerical parts of terms in algebraic expressions. They indicate how much the variable affects the term. Understanding coefficients helps you in predicting how changes in one part of an expression alter the whole expression.

In $x^2 + 6xy + 9y^2$,

The coefficient of $x^2$ is 1 (which is often implied, so it's not written).
The coefficient of $xy$ is 6, representing the combined effect of $x$ and $y$ in that term.
The coefficient of $y^2$ is 9.

Identifying coefficients is crucial for making comparisons or applying algebraic identities correctly.

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables within them. These identities simplify complex expressions into manageable parts or help in factoring expressions efficiently. Some famous identities include the difference of squares, the square of a sum, and the square of a difference.

For $x^2 + 6xy + 9y^2$, the identity used is the perfect square trinomial identity:

$(a+b)^2 = a^2 + 2ab + b^2$

Identifying when and how to use these identities can simplify solving problems and deepen understanding of algebraic structures.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Factor. $$x^{2}+6 x y+9 y^{2}$$

Short Answer

Step by step solution

Identify coefficients

Check if the equation is a perfect square trinomial

Apply the formula

Key Concepts

Perfect Square Trinomial

Quadratic Expression

Coefficients

Algebraic Identities

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Calculus

Statistics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.