/*! 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 62 \((x+6)(x+6)\)... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 62

\((x+6)(x+6)\)

Short Answer

Expert verified
The expanded form is \( x^2 + 12x + 36 \)

Step by step solution

01

- Identify the expression

Recognize that \( (x+6)(x+6) \) can be simplified because it is the product of two identical binomials.
02

- Apply the formula

Use the formula for squaring a binomial: \[ (a+b)^2 = a^2 + 2ab + b^2 \]. For the given expression, \( a \) is \( x \) and \( b \) is \( 6 \).
03

- Expand the binomial

Expand using the formula: \[ (x+6)^2 = x^2 + 2 \cdot x \cdot 6 + 6^2 \]
04

- Simplify the result

Finally, simplify the expression: \[ x^2 + 12x + 36 \]

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Key Concepts

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

Headline of the respective core concept
Squaring a binomial is a fundamental concept in algebra that involves multiplying a binomial by itself. Let's explore this concept by breaking down each key element involved in the process. The given exercise, \( (x + 6)(x + 6) \), serves as a perfect example to understand squaring binomials. To begin, we recognize that we are dealing with identical binomials. Next, we use the binomial square formula.
binomial expansion
When we talk about binomial expansion, we refer to the process of expanding an expression that is raised to a power. The general formula for expanding a squared binomial \[ (a + b)^2 \] is given by: \[(a + b)^2 = a^2 + 2ab + b^2\]. Here, the term \(2ab\) accounts for the product of the mixed terms. In our exercise, we apply the values \((x + 6)^2\), identifying \(a = x\) and \(b = 6\). This leads to: \[(x+6)^2 = x^2 + 2(x)(6) + 6^2\]. Understanding this formula is crucial.
algebraic expressions
An algebraic expression is a mathematical phrase involving numbers, variables, and operation symbols. In our exercise, \(x\) and \(6\) are the terms within the binomial making up the expression \( (x + 6) \). During expansion, each term within the binomial is considered for multiplication as per algebraic rules. Our aim is to simplify the expression to its most reduced form, which is achieved by performing multiplications and additions step by step.
simplification
Simplification involves reducing an expression to its simplest form. From our expanded binomial, \[(x + 6)^2 = x^2 + 12x + 36\], we simplify this by performing the arithmetic operations:
\(x^2\) remains as is since it's already in its simplest form.
The term \((2 \cdot x \cdot 6) \) simplifies to \(12x\).
Finally, \((6^2)\) is simplified as \((6 \cdot 6 = 36)\).
Hence, our final simplified expression is: \{ x^2 + 12x + 36 \}.
Simplifying expressions helps in solving mathematical problems more effectively and understanding deeper algebraic concepts.

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