/*! 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 34 For the following problems, find... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 34

For the following problems, find the products. $$ \left(a+\frac{1}{2}\right)^{2} $$

Short Answer

Expert verified
Answer: The product of the given expression is \(a^2 + a + \frac{1}{4}\).

Step by step solution

01

Identify the binomial expression

The given expression is a binomial of the form \(a+b\), where \(a=a\) and \(b=\frac{1}{2}\).
02

Apply the binomial expansion formula

We will use the formula \((a+b)^2 = a^2 + 2ab + b^2\) with our values of \(a\) and \(b\). Substituting our values of \(a\) and \(b\) into the formula, we get: $$ \left(a+\frac{1}{2}\right)^{2} = a^2 + 2\times a\times \frac{1}{2} + \left(\frac{1}{2}\right)^2 $$
03

Simplify the expression

Now, let's simplify the expression step by step: $$ = a^2 + a + \frac{1}{4} $$ So, the product of the given expression is: $$ \left(a+\frac{1}{2}\right)^{2} = a^2 + a + \frac{1}{4} $$

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
Algebra is a significant branch of mathematics that deals with symbols and the rules for manipulating these symbols. It provides a way to write formulas and solve equations using mathematical expressions. In our example, the expression \(\left(a+\frac{1}{2}\right)^{2}\) involves the use of variables and constants. Here, the variable is \(a\) and the constant is \(\frac{1}{2}\).
When working with algebra:
  • Understand that variables represent numbers and can change.
  • Apply arithmetic operations upon the variables and constants.
  • Utilize algebraic expressions to model real-life situations.
Algebra is essential as it forms the basis for more advanced topics, helping simplify complex problems, and offering solutions in a structured manner.
Binomial Theorem
The binomial theorem is a powerful tool in algebra, offering a way to expand expressions that are raised to a power. A binomial is a simple algebraic expression comprising two terms, like \(a + b\). The binomial theorem gives us a formula for expanding expressions like \((a + b)^n\). For smaller powers like \((a + b)^2\), it is a shortcut to avoid manual multiplication.
In our solution:
  • The formula used is \((a+b)^2 = a^2 + 2ab + b^2\) which simplifies the expansion process.
  • By substituting \(a\) and \(b\) from the expression \((a+\frac{1}{2})^2\), calculations become straightforward and efficient.
Understanding the binomial theorem allows us to manage complex algebraic expansions with ease, making it an essential concept for both basic and more advanced levels of study.
Polynomials
Polynomials are expressions involving a sum of powers of one or more variables multiplied by coefficients. They can be simple, like \(a\), or more complex, like \(a^2 + a + \frac{1}{4}\), where multiple terms are combined.
Some key aspects include:
  • Each term in a polynomial consists of a coefficient and a variable raised to a power.
  • Polynomials are classified by degree, which is the highest power of the variable in the expression.
  • In our given expression, \(a^2 + a + \frac{1}{4}\), the degree is 2, indicating a quadratic polynomial.
Polynomials are important in various fields such as physics, engineering, and economics, as they often represent relationships and trends. Recognizing and manipulating polynomial expressions is a fundamental skill in algebra, aiding in problem-solving and analytical thinking.

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

For the following problems, simplify each of the algebraic expressions. $$ x y(3 x y+2 x-5 y)-2 x^{2} y^{2}-5 x^{2} y+4 x y^{2} $$

For the following problems, simplify each of the algebraic expressions. $$ 3 h[2 h+5(h+2)] $$

Name the property of real numbers that makes \(4 y x^{2}=4 x^{2} y\) a true statement.

For the following problems, simplify each of the algebraic expressions. $$ 2(x-6)+5 $$

For the following problems, perform the multiplications and combine any like terms. $$ 6 a^{3} b^{3}\left(4 a^{2} b^{6}+7 a b^{8}+2 b^{10}+14\right) $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.