/*! 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 47 In Exercises \(45-62,\) multiply... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 47

In Exercises \(45-62,\) multiply using the rules for the square of a binomial. $$(2 x+5)^{2}$$

Short Answer

Expert verified
The short answer to the expression \((2x+5)^2\) is \(4x^2 + 20x + 25\).

Step by step solution

01

Identify \(a\) and \(b\)

Identify \(a\) and \(b\) based on given expression. In the expression \((2x+5)^2\), \(a\) can be considered as \(2x\) and \(b\) as \(5\).
02

Apply the formula

Use the formula of binomial square which is \((a+b)^2 = a^2 + 2ab + b^2\). Substitute the values \(a = 2x\) and \(b = 5\) and compute the value correctly.
03

Perform the operations

Simplify the equation after the operations have been applied. \( (2x)^2 + 2*(2x)*5 + (5)^2 = 4x^2 + 20x + 25 \)
04

Final Solution

The final solution is \(4x^2 + 20x + 25\) after performing all the operations in the previous steps.

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.

Algebraic Expressions
Algebraic expressions are a blend of numbers, variables, and mathematical operations. They form the foundational building blocks in algebra, allowing us to represent real-world problems symbolically. Think of them as mathematical phrases. For instance, in \((2x + 5)\), there are two terms:
  • \(2x\), where \(2\) is the coefficient and \(x\) is the variable.
  • \(5\), a constant term.
Understanding how to manipulate these expressions, such as expanding them or simplifying, is crucial. It sets the stage for solving more complex equations and is essential for learning the rules of polynomial multiplication, which we'll delve into next.
Polynomial Multiplication
Polynomial multiplication involves multiplying two or more polynomials together to form a new polynomial. It's an essential skill in algebra.

In the exercise \((2x + 5)^2\), we use the rule for squaring a binomial, which is a type of polynomial. The formula \((a+b)^2 = a^2 + 2ab + b^2\) is applied to simplify the process.
  • First, square the first term: \((2x)^2 = 4x^2\).
  • Then, calculate the double product: \(2 \times (2x) \times 5 = 20x\).
  • Finally, square the last term: \(5^2 = 25\).
When we combine these results, we get the new polynomial \(4x^2 + 20x + 25\). This technique helps in expanding expressions precisely and efficiently, which is key when tackling quadratic equations.
Quadratic Equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\) and play a crucial role in various areas of mathematics. They typically represent parabolic graphs and often appear in problems dealing with motion, area, and other physical phenomena.

In the exercise, the result \(4x^2 + 20x + 25\) can be viewed as a quadratic expression, though not set to zero. Identifying, solving, and graphing such equations involve:
  • Factoring the quadratic expression when possible.
  • Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • Recognizing that the solutions are the points where the graph of the equation intersects the x-axis.
Mastering quadratic equations empowers you to explore more complex algebraic concepts and equations. Embrace these skills to solve diverse mathematical challenges.

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

We have seen that in \(2009,\) the United States government spent more than it had collected in taxes, resulting in a budget deficit of \(\$ 1.35\) trillion. a. Express 1.35 trillion in scientific notation. b. A trip around the world at the Equator is approximately \(25,000\) miles. Express this number in scientific notation. c. Use your scientific notation answers from parts (a) and (b) to answer this question: How many times can you circle the world at the Equator by traveling 1.35 trillion miles?

Perform the indicated computations. Express answers in scientific notation. $$\frac{\left(1.2 \times 10^{6}\right)\left(8.7 \times 10^{-2}\right)}{\left(2.9 \times 10^{6}\right)\left(3 \times 10^{-3}\right)}$$

Use a vertical format to find each product. $$\begin{array}{r}2 x^{3}+x^{2}+2 x+3 \\\\\quad x+4 \\\\\hline\end{array}$$

Perform the indicated operations. $$(y+1)\left(y^{2}-y+1\right)+(y-1)\left(y^{2}+y+1\right)$$

Perform the indicated computations. Express answers in scientific notation. $$\left(5 \times 10^{3}\right)\left(1.2 \times 10^{-4}\right) \div\left(2.4 \times 10^{2}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

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.