Problem 50 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (3 x-\sqrt{2})(3 x-\sqrt{2}) $$

Step-by-Step Solution

Answer

The simplified expression is \(9x^2 - 6x\sqrt{2} + 2\).

Step 1: Recognize the Expression Format

This expression is a binomial multiplied by itself, often referred to as a binomial squared. Such an expression can be written in the form \((a - b)^2\), where \(a = 3x\) and \(b = \sqrt{2}\).

Step 2: Apply the Binomial Square Formula

To expand the expression \((a - b)^2\), we use the formula \(a^2 - 2ab + b^2\). Following this formula, we expand our expression:1. \(a^2 = (3x)^2\)2. \(-2ab = -2(3x)(\sqrt{2})\)3. \(b^2 = (\sqrt{2})^2\).

Step 3: Calculate Each Term

Let's calculate each part of the expression:- Calculate \((3x)^2 = 9x^2\).- Calculate \(-2(3x)(\sqrt{2}) = -6x\sqrt{2}\).- Calculate \((\sqrt{2})^2 = 2\).

Step 4: Combine the Terms

Add all the expanded terms together:- \(9x^2 - 6x\sqrt{2} + 2\).This is the fully expanded and simplified expression.

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that helps us expand expressions raised to a power. A binomial is an algebraic expression containing two terms, like \((a - b)^n\). In our exercise, we dealt with a binomial squared, which simplifies using the binomial expansion formula for \((a - b)^2\): \[ (a - b)^2 = a^2 - 2ab + b^2 \].
This is a special case of the binomial theorem that helps us tackle binomials squared efficiently. When expanding binomials in this way:

• Apply the formula by identifying your \(a\) and \(b\) terms.
• Square the first term to get \(a^2\).
• Multiply both terms by 2 and then by each other to get \(-2ab\).
• Square the second term to get \(b^2\).

Using this straightforward method allows us to expand and simplify these types of expressions systematically.

Algebraic Expressions

Algebraic expressions are mathematical phrases that encompass numbers, variables, and operations. They form the cornerstones of algebra, allowing us to articulate complex mathematical ideas in simpler terms. In the context of our exercise, the expression \((3x - \sqrt{2})(3x - \sqrt{2})\) is an algebraic expression made of two terms: \(3x\) and \(\sqrt{2}\).
To comprehend and manipulate algebraic expressions:

• Identify all the variables and constants involved.
• Recognize operations like addition, subtraction, multiplication, and division, which join the components of the expression.
• Rewriting the expression might involve expanding or factoring, depending on the requirement.

Being adept at handling algebraic expressions enables solving equations, simplifying problems, and attaining solutions more effectively.

Simplification in Algebra

Simplification in algebra revolves around reducing expressions to their most fundamental form. This can involve combining like terms, using properties of operations, and reducing complexity to make equations easier to solve or understand. In our original exercise, simplification followed after expanding the terms using the binomial expansion.
Key steps in simplifying an algebraic expression include:

• Combine like terms, which are terms with the same variables raised to the same power, such as \(9x^2\) in our example.
• Simplify radicals, like \((\sqrt{2})^2\) which equals \2\.
• Reorder or structure terms for clarity, ensuring the expression is both accurate and concise.

The final expression, \(9x^2 - 6x\sqrt{2} + 2\), is considered simplified because it cannot be reduced further. Simplified expressions often make it easier to interpret results and perform further mathematical operations if needed.