Problem 65 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{3}+x)^{2} $$

Step-by-Step Solution

Answer

The result is \(3 + 2\sqrt{3}x + x^2\).

Step 1: Understand the Binomial Expression

We have the binomial expression \((\sqrt{3} + x)^2\) which means it should be expanded using the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = \sqrt{3}\) and \(b = x\).

Step 2: Square the First Term

In the expression \((\sqrt{3} + x)^2\), the first term is \(\sqrt{3}\). Calculating \((\sqrt{3})^2\), we get \(3\). Hence \(a^2 = 3\).

Step 3: Calculate Twice the Product of the Two Terms

Now calculate \(2ab\), which is twice the product of \(\sqrt{3}\) and \(x\). So, \(2 \cdot \sqrt{3} \cdot x = 2\sqrt{3}x\).

Step 4: Square the Second Term

For the term \(x\), we have \(x^2\) which is simply \(x^2\).

Step 5: Combine All Parts

Combine all the calculated parts from steps 2, 3, and 4: \(3 + 2\sqrt{3}x + x^2\). This is the expanded form of the expression that cannot be simplified further.

Key Concepts

Square of a Binomial

Understanding the square of a binomial is essential in many areas of algebra. A binomial is simply an expression with two terms, like \(a + b\). Squaring a binomial involves multiplying it by itself. The formula used is

• \((a + b)^2 = a^2 + 2ab + b^2\)

This formula efficiently expands the binomial, making calculations simple. Let's break it down:

• \(a^2\) is the square of the first term.
• \(2ab\) is twice the product of the two terms.
• \(b^2\) is the square of the second term.

For the expression \((\sqrt{3} + x)^2\), let's identify our terms:

• \(a = \sqrt{3}\), meaning \(a^2 = 3\).
• \(b = x\), resulting in \(b^2 = x^2\).
• The middle term becomes \(2 \cdot \sqrt{3} \cdot x = 2\sqrt{3}x\).

Thus, the expanded form becomes \(3 + 2\sqrt{3}x + x^2\). This clear structure helps in subsequent simplification and understanding.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about making them easier to understand and work with. The goal is to arrange terms to see the relationships and operations clearly. For instance, when expanding \((\sqrt{3} + x)^2\), it isn't just about multiplying;

• It's about applying the formula \((a + b)^2 = a^2 + 2ab + b^2\) correctly.
• Then combining similar terms, if possible, to create a streamlined expression.

In our example, each term - \(3\), \(2\sqrt{3}x\), \(x^2\) - remains distinct without further simplification. The result \(3 + 2\sqrt{3}x + x^2\) is as simple as it gets because:

• There are no like terms to combine further.
• Each component is unique, with its own algebraic identity.

This process helps solidify understanding and lays the groundwork for more complex operations.

Positive Real Numbers

When dealing with algebraic expressions, particularly those involving variables and radicals, we often assume variables represent positive real numbers. This assumption ensures that all operations, especially involving radicals like square roots, are valid and meaningful.

• Radicals, such as the square root, are inherently defined for non-negative numbers.
• Positive real numbers make calculations straightforward, avoiding undefined or complex outputs.

In our expression \((\sqrt{3} + x)^2\), considering \(x\) as a positive real number is key:

• The expression \(\sqrt{3}\) is already a positive real number, valued approximately at 1.732.
• \(x\) should be positive to complement and maintain this property throughout simplification.

This premise resolves potential ambiguities, making the expansion and simplification of algebraic expressions smoother and more predictable. Understanding this is vital as it aids in accurate mathematical communication and analysis.