Problem 54 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ (2-\sqrt{6})^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((2-\sqrt{6})^2\). Answer: The simplified expression is \((2 - \sqrt{6})^2 = 10 - 4\sqrt{6}\).

Step 1: Identify the binomial#a-b

In this case, the binomial is \((2-\sqrt{6})\). We have that \(a = 2\) and \(b = \sqrt{6}\).

Step 2: Apply the binomial theorem formula

We will use the formula: \((a - b)^2 = a^2 - 2ab + b^2\). Substitute \(a = 2\) and \(b = \sqrt{6}\) into this formula: \((2 - \sqrt{6})^2 = 2^2 - 2(2)(\sqrt{6}) + (\sqrt{6})^2\)

Step 3: Compute the squares and products

Calculate the squares and products in the expression: \(2^2 = 4\) \(-2(2)(\sqrt{6}) = -4\sqrt{6}\) \((\sqrt{6})^2 = 6\)

Step 4: Substitute values back into the expression and simplify

We substitute the computed values back into the expression and simplify: \((2 - \sqrt{6})^2 = 4 - 4\sqrt{6} + 6\) Now, simplify the expression by combining like terms: \((2 - \sqrt{6})^2 = 4 + 6 - 4\sqrt{6} = 10 - 4\sqrt{6}\) The simplified expression is: \((2 - \sqrt{6})^2 = 10 - 4\sqrt{6}\).

Key Concepts

Algebraic Expressions

In mathematics, an algebraic expression is a combination of numbers, variables, and operations like addition or multiplication. These expressions can vary from simple to complex. They form the basis of algebra, which uses symbols to represent numbers.

Key components of algebraic expressions:

• Variables: Symbols, usually letters like \(x\) or \(y\), that stand for unknown values.
• Constants: Numbers on their own that have a fixed value.
• Operators: Symbols that indicate an operation to be performed, such as +, -, *, and \(\div\).

Algebraic expressions allow us to formulate mathematical phrases that can be simplified or expanded using various methods. These expressions help solve equations and model real-world scenarios, becoming a powerful tool to describe patterns and solve problems.

Square of a Binomial

When we encounter the term 'square of a binomial,' it refers to the product of a binomial with itself. A binomial is an algebraic expression with two terms. Examples include

• \((x + y)\)
• \((3 - 5)\)

To square a binomial like \((a - b)\), you apply the formula: \[(a - b)^2 = a^2 - 2ab + b^2\]

This formula is derived from multiplying \((a-b)\) by \((a-b)\), which expands into three key terms:

• The square of the first term: \(a^2\)
• Twice the product of both terms: \(-2ab\)
• The square of the second term: \(b^2\)

Understanding this pattern helps quickly simplify expressions like \((2 - \sqrt{6})^2\), leading to correct solutions without lengthy calculations.

Simplifying Expressions

Simplifying expressions is the process of rewriting them in their simplest form. This involves reducing an expression to as few terms as possible and eliminating common factors. It's important in making expressions easier to understand and solve.

Steps involved in simplifying expressions:

• Perform Operations: Carry out any indicated operations in the expression.
• Combine Like Terms: Group and merge terms that have the same variables and powers.
• Reduce Complex Fractions: Simplify fractions to their lowest terms.

For example, when simplifying \((2 - \sqrt{6})^2\), after expanding using the binomial theorem, combine like terms to reach the final answer: \(10 - 4\sqrt{6}\).

Simplification is useful in algebraic manipulation, enabling us to solve or evaluate expressions more efficiently.