Problem 53 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \(3x + 6\sqrt{x} + 3\).

Step 1: Understand the Problem

We are given the expression \((\sqrt{3x} + \sqrt{3})^2\) and we need to expand and simplify it.

Step 2: Recognize the Pattern

Notice that this expression takes the form \((a + b)^2\), which can be expanded using the identity \((a + b)^2 = a^2 + 2ab + b^2\).

Step 3: Identify Values for 'a' and 'b'

In our expression, \(a = \sqrt{3x}\) and \(b = \sqrt{3}\). We will use these values to expand the expression.

Step 4: Expand the Expression

Substitute \(a = \sqrt{3x}\) and \(b = \sqrt{3}\) into the identity \((a + b)^2 = a^2 + 2ab + b^2\). This gives us the expression: \[ a^2 + 2ab + b^2 = (\sqrt{3x})^2 + 2(\sqrt{3x})(\sqrt{3}) + (\sqrt{3})^2 \]

Step 5: Simplify Each Term

Now, compute each component individually:- \((\sqrt{3x})^2 = 3x\) - \(2(\sqrt{3x})(\sqrt{3}) = 2\sqrt{9x} = 6\sqrt{x}\)- \((\sqrt{3})^2 = 3\)

Step 6: Combine and Simplify

Combine the simplified terms:\[ 3x + 6\sqrt{x} + 3 \] This is the simplified form of the original expression.

Key Concepts

Expanding Expressions

Expanding algebraic expressions is a crucial concept, as it allows us to convert expressions from a compact form to a detailed equivalent. This process often utilizes algebraic identities, making it easier to understand and manage the expressions.
For our given expression, \((\sqrt{3x} + \sqrt{3})^2\), expanding it means we'll employ the identity \((a+b)^2 = a^2 + 2ab + b^2\). This identity helps us distribute the squared term, breaking it down into simpler components that can be computed separately.

• The expression is in the form of \((a + b)^2\) with \(a = \sqrt{3x}\) and \(b = \sqrt{3}\).
• By expanding, we compute each term step by step: \(a^2\), \(2ab\), and \(b^2\).
• This approach doesn't just apply to binomials; it can be expanded to other forms, helping to address complex algebraic expressions with multiple terms.

Whether dealing with simple binomials or more complex polynomials, mastering expansion techniques simplifies solving algebraic problems.

Square of Binomial

The square of a binomial is a frequently encountered concept in algebra, particularly when we want to evaluate expressions like \((a + b)^2\). This squaring process applies a specific formula to efficiently break down the expression.
In the expression \((\sqrt{3x} + \sqrt{3})^2\), each part of the formula plays a role:

• \(a^2\) calculates the square of the first term.
• \(2ab\) computes the product of twice both terms, capturing the interaction between them.
• \(b^2\) finds the square of the second term.

This formula becomes second nature as you dive deeper into algebra, allowing for quick and accurate expansion of any binomial squared expression.
Through this approach, you can also understand how products and powers interact, ensuring a thorough grounding in fundamental algebraic techniques.

Simplification

Simplification is a powerful tool in algebra that transforms complex expressions into their most straightforward form. By simplifying, we reduce terms and expressions, making them easier to understand and work with.
In our expression, after applying the binomial square formula to \((\sqrt{3x} + \sqrt{3})^2\), we reach an intermediate form:

• \((\sqrt{3x})^2 = 3x\)
• \(2(\sqrt{3x})(\sqrt{3}) = 6\sqrt{x}\)
• \((\sqrt{3})^2 = 3\)

By combining these simplified terms, \[ 3x + 6\sqrt{x} + 3 \]we arrive at a simplified expression. Simplification involves recognizing opportunities to rearrange and combine like terms.
This process is not only useful in making computations manageable but also essential in solving equations, and inequalities, and evaluating functions efficiently. Understanding simplification will greatly enhance problem-solving skills in various mathematical problems.