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Magazine Chapter 5: Problem 66
Multiply. See Examples I through II. $$ (4 x+6)^{2} $$
Short Answer
Expert verified
The expanded form is \(16x^2 + 48x + 36\).
Step by step solution
01
Understand the Problem
Our task is to expand the expression \((4x + 6)^2\). This is in the form of \((a+b)^2\), which follows the expansion formula \((a+b)^2 = a^2 + 2ab + b^2\).
02
Identify Components
In the expression \((4x + 6)^2\), identify \(a\) and \(b\). Here, \(a = 4x\) and \(b = 6\).
03
Apply the Formula
Using the formula \((a+b)^2 = a^2 + 2ab + b^2\), substitute \(a = 4x\) and \(b = 6\) into the formula.
04
Calculate \(a^2\)
Compute \((4x)^2\), which gives \(16x^2\).
05
Calculate \(2ab\)
Compute \(2 \times 4x \times 6\), which results in \(48x\).
06
Calculate \(b^2\)
Compute \(6^2\), which gives 36.
07
Combine the Results
Combine all parts to write the expanded form: \(16x^2 + 48x + 36\).
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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 provides a shortcut method for expanding expressions that are raised to a power, such as \((a+b)^n\). A binomial is an algebraic expression that contains two distinct terms which are often written within parentheses. When you see a binomial raised to a power, the Binomial Theorem can be used to simplify the expression without doing the calculations manually for each power.
- When multiplying a binomial, you rely on a pattern to determine each term in the expansion.
- The expression \((a+b)^2\) serves as a basic introduction to the binomial expansion because of its simplicity.
Algebraic Expressions
An algebraic expression is a combination of variables, constants, and arithmetic operations.In the expression \((4x + 6)^2\), we observe two components, "4x" and "6".
The variables represent unknown quantities, while the constants are fixed numbers whose values do not change.
- "4x" indicates a variable term where "4" is a coefficient and "x" is a variable.
- "6" is a constant term, representing a fixed number.
The variables represent unknown quantities, while the constants are fixed numbers whose values do not change.
Exponents
Exponents are used to denote repeated multiplication of a base number.In the expression \((4x + 6)^2\), the "2" is an exponent, indicating that the base binomial, \(4x+6\), is multiplied by itself once.
- Exponents simplify expressions by showing how many times to use the base as a factor.
- When expanding using the base as a binomial, each term's degree is determined by the exponent.
Quadratic Expressions
Quadratic expressions involve terms where the variable is raised to an exponent of 2, resulting in a polynomial of degree 2.The expanded form of our binomial, \(16x^2 + 48x + 36\), exemplifies a quadratic expression.
- The term \(16x^2\) is the quadratic term, as it contains \(x\) raised to the second power.
- \(48x\) denotes the linear term, involving \(x\) raised to the first power.
- \(36\) is the constant term, independent of \(x\).
