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Chapter 5: Problem 10

Multiply. $$(a+2)(a+2)$$

Short Answer

Expert verified
\[(a+2)^2 = a^2 + 4a + 4\]

Step by step solution

01

Recognize the expression

Identify the mathematical operation required. The given expression is \( (a+2)(a+2) \). This is a binomial being multiplied by itself.
02

Apply the formula for square of a binomial

Use the formula \( (x+y)^2 = x^2 + 2xy + y^2 \). In this case, x is \(a\) and y is \(2\).
03

Substitute the values

Substitute x \(= a\) and y \(= 2\) into the formula \[(a+2)^2 = a^2 + 2(a)(2) + (2)^2\].
04

Simplify the expression

Calculate each term: \(a^2\) remains as is, \(2(a)(2)\) becomes \(4a\), and \(2^2\) simplifies to \(4\).
05

Combine all terms

Add the simplified terms: \[(a+2)^2 = a^2 + 4a + 4\].

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Key Concepts

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

Binomial Multiplication
To understand binomial multiplication, let's break down what a binomial is.
A binomial is an algebraic expression with two terms, like \(a+2\) or \(x-5\).
Multiplying binomials involves using the distributive property to ensure every term in the first binomial is multiplied by every term in the second binomial.
In the given exercise, we have \((a+2)(a+2)\), which is a binomial multiplied by itself. This is also called squaring a binomial.
It can be simplified step by step using the square of a binomial formula.
Algebraic Expressions
Algebraic expressions combine numbers, variables, and arithmetic operations.
They include entities like constants, coefficients, and variables.
For example, in the expression \(3a + 4\), \3\ is the coefficient and \a\ is the variable. The expression also contains a constant term \4\.
In our problem \((a+2)(a+2)\), \a\ is the variable, and \2\ is a constant.
Understanding algebraic expressions is key to solving and simplifying them effectively.
Always remember to perform operations according to the algebraic rules.
Expansion Formula
The expansion formula for the square of a binomial is \((x+y)^2 = x^2 + 2xy + y^2\).
This formula helps in expanding and simplifying expressions where a binomial is squared.
Using our given problem \((a+2)^2\), we identify \a\ as \x\ and \2\ as \y\.
By substituting these values, we get \a^2 + 2(a)(2) + 2^2\.
Simplifying this, we get the terms \a^2, 4a,\ and \4\.
Combining these, we finally get the expanded form \a^2 + 4a + 4\.
This expansion formula is essential for dealing with polynomials and higher-level math problems.

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