/*! 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 29 Expand each expression. $$ (... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 29

Expand each expression. $$ (g+7)^{2} $$

Short Answer

Expert verified
The expanded form is \( g^2 + 14g + 49 \).

Step by step solution

01

Identify the binomial

Recognize that the expression \( (g+7)^{2} \) is a binomial raised to a power of 2.
02

Apply the Binomial Theorem

The binomial theorem states that \( (a+b)^2 = a^2 + 2ab + b^2 \). Here, \( a = g \) and \( b = 7 \).
03

Square the first term

Calculate \( g^2 \). This gives \( g^2 \).
04

Multiply the two terms and then by 2

Multiply \( g \) and \( 7 \) and then multiply the result by 2: \( 2 \times g \times 7 = 14g \).
05

Square the second term

Calculate \( 7^2 \). This gives \( 49 \).
06

Combine all terms

Combine all the terms: \( g^2 + 14g + 49 \).

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Key Concepts

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

binomial theorem
The binomial theorem is a fundamental principle in algebra. It provides a formula for expanding expressions that are raised to any power. Specifically, for any binomial expression \textcolor{red}{\((a + b)^n\)}, it can be expanded into a sum of terms of the form \textcolor{red}{\(C(n, k) a^{n-k} b^k\)}. This means, when we expand \textcolor{red}{\((g+7)^2\)}, we can break it down into \textcolor{red}{\((a+b)^2\)}, where \textcolor{red}{\(a=g\)} and \textcolor{red}{\(b=7\)}. The binomial theorem helps us quickly find the expanded form as \textcolor{red}{\(a^2 + 2ab + b^2\)}.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication). They form the cornerstone of algebra. In the exercise, \textcolor{red}{\((g+7)^2\)} is an algebraic expression involving the variable \textcolor{red}{\(g\)} and the constant \textcolor{red}{\(7\)}. When expanding such expressions, our goal is to rewrite them in a more simplified or detailed form. Recognizing patterns in algebraic expressions, like the binomial form \textcolor{red}{\((a+b)\)}, helps in manipulating and understanding more complex algebraic problems.
squaring binomials
Squaring binomials is a specific application of the binomial theorem. When we square a binomial like \textcolor{red}{\((g+7)\)}, we follow the formula \textcolor{blue}{ \((a+b)^2 = a^2 + 2ab + b^2\) }. Here's a simple breakdown:

  • First, square the first term: \textcolor{red}{\(g^2\)}

  • Then, multiply both terms together and then by two: \textcolor{red}{14g}

  • Finally, square the second term: \textcolor{red}{\(49\)}
Combining these results gives us the expanded form \textcolor{blue}{ \(g^2 + 14g + 49\) }. This method simplifies and standardizes how we deal with binomials, making it easier to handle in algebraic equations and expressions.

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Most popular questions from this chapter

Rewrite each expression as a square with a constant added or subtracted. $$ p^{2}-16 p+60 $$

Challenge Although these equations are not quadratic, the quadratic formula can help you solve them. Try to solve them, and explain your reasoning. $$ \left(x^{2}-2 x-2\right)^{2}=0 $$

Many equations cannot be solved directly by backtracking. Some have the variable stated more than once; others involve variable as exponents. Here are some equations that can't be solved directly by backtracking. $$ \begin{array}{c}{f^{2}=f+1} & {x=\sqrt{x}+1 \quad k^{2}+k=0} \\ {1.1^{B}=2} & {\frac{1}{x}=x^{2}+2}\end{array} $$ For each equation below, write yes if it can be solved directly by backtracking and \(n o\) if it cannot. $$ \begin{array}{ll}{\text { a. } 5=\sqrt{x-11}} & {\text { b. } 4^{d}=9} \\\ {\text { c. } 3 g^{2}=5} & {\text { d. } \sqrt{x+1}=x-4}\end{array} $$

Solve each equation. Be sure to check your answers. (Hint: Try factoring the numerator first.) $$ \frac{16 x^{2}-81}{4 x+9}=31 $$

Factor each quadratic expression that can be factored using integers. Identify those that cannot, and explain why they can't be factored. $$ z^{2}+2 z-6 $$
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