Problem 54 - Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (\sqrt{5 x}-\sqrt{3})^{2} $$

Step-by-Step Solution

Answer

The simplified result is \(5x + 3 - 2 \sqrt{15x}\).

Step 1: Apply the Binomial Square Formula

To square the binomial \((\sqrt{5x} - \sqrt{3})\), we use the formula \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = \sqrt{5x}\) and \(b = \sqrt{3}\).

Step 2: Square Each Term

First, square the terms \(a\) and \(b\):\[a^2 = (\sqrt{5x})^2 = 5x\]\[b^2 = (\sqrt{3})^2 = 3\]

Step 3: Calculate the Middle Term

The middle term in the binomial expansion is given by \(-2ab\). Substitute \(aed \)b with their respective values:\[-2(\sqrt{5x})(\sqrt{3}) = -2 \sqrt{15x}\]

Step 4: Write the Expanded Expression

Combine all the terms to write the expanded expression:\[5x - 2 \sqrt{15x} + 3\]

Step 5: Simplify the Expression

The expression is already simplified with terms combined properly:\[5x + 3 - 2\sqrt{15x}\]

Key Concepts

Binomial Square Formula

When dealing with algebraic expressions, especially those involving binomials, the Binomial Square Formula is a handy tool. This formula allows us to expand individual terms while capturing the interactions between them. For any binomial of the form \((a - b)^2\), the expansion is given by:

• \(a^2\): Square the first term.
• \(-2ab\): Multiply the two terms together, multiply the result by 2, and also include a negative sign. It accounts for the interaction between the two terms.
• \(b^2\): Square the second term.

Each component adds to the final expanded form, allowing us to move from a compact binomial to a more detailed expression. It’s important to recognize the components and apply them correctly to simplify complex algebraic problems.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This simplification helps in understanding and using expressions more effectively, especially when they involve radicals or multiple terms.

Key points to consider while simplifying include:

• Combine like terms: Terms that have the same variable components can be added or subtracted. For example, if the expression has two \(x\) terms, they should be combined into one.
• Follow the order of operations: Ensure the calculations, such as multiplication before addition, are carried out as per the standard math rules.
• Simplify radicals when possible: If you have radicals like \(\sqrt{15x}\), ensure it can’t be simplified further. In some cases, they might be simplified by extracting factors that are perfect squares.

Once all these steps are applied, expressions become more straightforward and easier to understand, aiding further algebraic manipulations.

Radicals in Algebra

Radicals, often represented by the square root or nth roots, are an essential part of algebra. They enable representation of expressions that involve square roots or cube roots, which appear quite frequently in math problems.

Understanding radicals involves:

• Recognizing radicals: A radical can always be represented with the root sign, like \(\sqrt{\cdot}\), for square roots, or with an index, such as \(\sqrt[3]{\cdot}\) for cube roots.
• Manipulating radicals: Radicals can be multiplied and divided, but caution is needed especially when adding or subtracting. Ensure all radicals are simplified first.
• Simplifying radicals: If a number inside a radical is a perfect square (like 4 within \(\sqrt{4}\)), the radical can be simplified directly to its integer form.

Dealing with radicals properly is crucial because they often contribute to the complexity of algebraic expressions. Ensuring they are properly simplified and correctly handled makes solving algebraic problems much smoother.