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Chapter 1: Problem 8

Perform the indicated operation and simplify. $$(b-7)^{2}$$

Short Answer

Expert verified
The simplified form is \ b^{2} - 14b + 49 \.

Step by step solution

01

- Write the expression

The given expression is \( (b-7)^{2} \)
02

- Apply the Square of a Binomial Formula

Use the formula \[ (a-b)^{2} = a^{2} - 2ab + b^{2} \] where \( a = b \) and \( b = 7 \)
03

- Substitute values into the formula

Substitute \( a = b \) and \( b = 7 \) into the formula: \[ (b-7)^{2} = b^{2} - 2(b)(7) + 7^{2} \]
04

- Simplify each term

Calculate and simplify each term: \[ b^{2} - 2(b)(7) + 7^{2} = b^{2} - 14b + 49 \]
05

- Combine all the terms

Combine the simplified terms to get the final expression: \[ b^{2} - 14b + 49 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Formula
In algebra, a binomial is an expression containing two terms, usually connected by a plus or minus sign. The binomial formula, specifically, helps us expand expressions that are raised to a power. For example, \( (a - b)^{2} \). The formula for this is \[ (a - b)^{2} = a^{2} - 2ab + b^{2} \]. This formula helps us break down and simplify expressions involving binomials. It's very handy as it saves us from multiplying binomials manually, making complex problems much easier to solve.
Algebraic Expressions
An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. In our exercise, \( (b-7)^{2} \), this is an algebraic expression where \( b \) is our variable, and \( 7 \) is a constant. Algebraic expressions can be simplified using different algebraic rules and formulas. Understanding how to manipulate these expressions is fundamental in algebra, as it allows you to solve equations and understand relationships between variables and constants. Mastering this will make working with algebraic problems much more manageable.
Simplification
Simplification in algebra means to make an expression easier to read and work with. It involves combining like terms and applying mathematical operations to present the expression in its simplest form. In our example, to simplify \( (b - 7)^{2}\), we applied the binomial formula to expand it and then combined like terms. This gave us \[ b^{2} - 14b + 49 \]. Simplification helps in solving algebraic equations and understanding the expressions better by reducing them to their most basic components.
Here are some tips for simplification:
  • Combine like terms.
  • Apply algebraic formulas.
  • Factor when possible.

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Most popular questions from this chapter

a. Write an absolute value equation or inequality to represent each statement. b. Solve the equation or inequality. Write the solution set to the inequalities in interval notation. The value of \(x\) differs from 4 by more than 1 unit.

Suppose that over the course of 3 months, a court system selects 1000 jurors from a large jury pool having an equal number of males and females. If males and females are selected at random, there should be approximately 500 males and 500 females selected. However, these values may vary slightly. If 1000 people are selected at random from the large jury pool, the inequality \(\left|\frac{x-500}{\sqrt{250}}\right|<1.96\) gives the "reasonable" range for the number of women selected, \(x\). a. Solve the inequality and interpret the answer in the context of this problem. (Hint: Round so that the endpoints of the interval are whole numbers, but still within the interval defined by the inequality.) b. If the group of 1000 jurors has 560 women, does it appear that there may be some bias toward women jurors?

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