/*! 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 46 Determine whether each of the fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 46

Determine whether each of the following is a difference of squares. $$ -12+25 t^{2} $$

Short Answer

Expert verified
No, \[-12 + 25t^2\] is not a difference of squares.

Step by step solution

01

Identify the general form of a difference of squares

The general form of a difference of squares is given by the equation \[a^2 - b^2 = (a + b)(a - b)\]. This form consists of two squared terms with a subtraction in between.
02

Compare the given expression with the difference of squares form

The given expression is \[-12 + 25t^2\]. Rewriting it in a more familiar form: \[25t^2 - 12\].
03

Check if both terms are perfect squares

The term \[25t^2\] is a perfect square because it can be written as \[(5t)^2\]. However, the term \[12\] is not a perfect square because there is no integer \(n\) such that \(n^2 = 12\).
04

Conclusion

Since \[12\] is not a perfect square, \[25t^2 - 12\] is not in the form of a difference of two squares.

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.

algebra
To understand this exercise, we need a basic grasp of algebra. Algebra allows us to work with mathematical symbols and manipulate them according to specific rules. In this case, we are using algebraic techniques to identify whether an expression fits a specific pattern known as the 'difference of squares'.

One essential algebraic pattern is the difference of squares, which involves expressions of the form \(a^2 - b^2\). Recognizing these patterns helps simplify and solve equations.

This exercise demonstrates the power of algebra by breaking down expressions into familiar forms. By rearranging the terms, we can see if they match the general pattern of a difference of squares.
perfect squares
Let's explore what perfect squares are before identifying them in expressions. A perfect square is an integer that can be written as the square of another integer. For example, \(1, 4, 9, 16\), and \(25\) are perfect squares because they are \(1^2, 2^2, 3^2, 4^2\), and \(5^2\) respectively.

In the given exercise, we examine whether each term in the expression \(25t^2 - 12\) is a perfect square. The term \(25t^2\) can be written as \((5t)^2\), which is a perfect square. However, \(12\) is not a perfect square because no integer squared equals \(12\). Hence, identifying perfect squares is crucial in classifying the difference of squares.
factoring
Factoring involves breaking down an expression into simpler terms that can be multiplied together to get the original expression. It's a handy tool in algebra for simplifying expressions and solving equations.

For a difference of squares, the factored form is given by \(a^2 - b^2 = (a + b)(a - b)\). This formula comes from the identity that any square minus another square equals the product of their sum and their difference.

In our exercise, we needed to see if the given expression \(25t^2 - 12\) could be factored using this pattern. We found that it couldn't because \(12\) is not a perfect square. Thus, factoring techniques rely on recognizing these perfect squares and patterns.

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 completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ a^{3}-a b^{2}-2 a^{2}+2 b^{2} $$

Review multiplying binomials using FOIL $$ (x-2)(x-7) $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ (p+q)^{2}-25 $$

Review multiplying binomials using FOIL $$ (t-5)(t+10) $$

Ashlee factors \(8 x^{2} y-10 x y^{2}\) as $$ 2 x y \cdot 4 x-2 x y \cdot 5 y $$ Is this the factorization of the polynomial? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.