/*! 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 2 A student incorrectly squared \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 2

A student incorrectly squared \((x-y)\) as follows. $$ (x-y)^{2}=x^{2}+y^{2} \quad $$

Short Answer

Expert verified
The correct result should be (x-y)^2 = x^2 - 2xy + y^2.

Step by step solution

01

- Identify Correct Formula for Squaring a Binomial

The correct formula for squaring a binomial is (a-b)^2 = a^2 - 2ab + b^2.
02

- Apply the Correct Formula to (x-y)^{2}

Using the formula from Step 1, we replace a with x and b with y. This gives (x-y)^2 = x^2 - 2xy + y^2.
03

- Compare the Incorrect and Correct Results

The incorrect result given by the student was x^2 + y^2. The correct result is x^2 - 2xy + y^2. The student missed the middle term -2xy and mistakenly added the squares of x and y directly.

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.

Binomial Theorem
The binomial theorem is a key concept in algebra that provides a formula for expanding the powers of a binomial. A binomial is an algebraic expression that contains exactly two terms, such as \(a + b\) or \(x - y\). Applying the binomial theorem helps us to expand these expressions without multiplying them out manually, which can save a lot of time and effort.
The expansion of \(a - b\) squared is a common application. According to the binomial theorem, we expand it using the formula:
\[ (a-b)^2 = a^2 - 2ab + b^2 \] This tells us that when squaring a binomial, we must:
  • Square the first term \(a\)
  • Square the second term \(b\)
  • Include twice the product of the two terms, with the appropriate sign based on the binomial
This understanding is essential for correctly handling binomials and avoiding common mistakes.
Quadratic Expressions
A quadratic expression is any polynomial of the form \(ax^2 + bx + c\), where \(a, b\,, and \ c\) represent constants. Squaring a binomial often results in a quadratic expression. For example, squaring \(x - y\) correctly results in the quadratic expression \(x^2 - 2xy + y^2\).
Quadratic expressions are important in algebra because they frequently appear in equations that need to be solved. Understanding how to derive and manipulate these expressions can help in various mathematical and practical applications.
Here are some points to remember:
  • The highest degree of a term (power of \(x\)) in a quadratic expression is 2.
  • The expression may contain a linear term (\bm{bx}) and a constant term (\bm{c}).
  • When expanding binomials into quadratic expressions, the accuracy of each term matters.
Always double-check your work to avoid leaving out necessary terms like the middle \(-2xy\) in our example.
Algebraic Mistakes
Algebraic mistakes are common, especially when working with more complex expressions like binomials and quadratic equations. Let鈥檚 address the mistake made in the exercise: the student squared \((x-y)^2\) incorrectly by writing it as \(x^2 + y^2\), instead of \(x^2 - 2xy + y^2\).
Common algebraic mistakes when handling binomials include:
  • Incorrectly applying binomial expansion formulas
  • Omitting necessary terms, such as the \(-2xy\) term
  • Adding or subtracting terms incorrectly
To minimize these errors:
  • Always write down the correct formula first.
  • Substitute the relevant terms carefully.
  • Verify each step by reapplying the concepts and formulas.
Remember, double-checking and understanding why the formula works can prevent most errors and ensure correct solutions.

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

The Double Helix Nebula, a conglomeration of dust and gas stretching across the center of the Milky Way galaxy, is 25,000 light-years from Earth. If one light-year is about \(6,000,000,000,000 \mathrm{mi},\) about how many miles is the Double Helix Nebula from Earth? (Data from www.spitzer.caltech.edu)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ \left(2 x^{2}+\frac{2}{3} y\right)\left(3 x^{2}-\frac{3}{4} y\right) $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 20 \frac{1}{2} \times 19 \frac{1}{2} $$

Find each product. $$ 2 x^{2}(2 x-5)(x+3) $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ \left(3 p^{2}+\frac{5}{4} q\right)\left(2 p^{2}-\frac{5}{3} q\right) $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure 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.