Problem 65 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\((\sqrt{3}+x)^2 = 3 + 2\sqrt{3}x + x^2\).

Step 1: Understand the Expression

The given expression is \((\sqrt{3} + x)^2\), which is a binomial raised to the power of two. We need to use the formula for the square of a sum: \((a+b)^2 = a^2 + 2ab + b^2\). In this case, \(a = \sqrt{3}\) and \(b = x\).

Step 2: Apply the Formula

Using the formula \((a+b)^2 = a^2 + 2ab + b^2\), substitute \(a=\sqrt{3}\) and \(b=x\). The expression becomes: \(a^2 = (\sqrt{3})^2 = 3\), \(2ab = 2(\sqrt{3})(x) = 2\sqrt{3}x\), and \(b^2 = x^2\).

Step 3: Combine the Terms

Combine the results from Step 2 to form the squared expression:\[3 + 2\sqrt{3}x + x^2\].

Step 4: Simplify (if possible)

The expression \(3 + 2\sqrt{3}x + x^2\) is already simplified because there are no like terms to combine. Each part of the expression is distinct.

Key Concepts

Binomial

A binomial expression is an algebraic expression that contains exactly two terms. These terms are connected by either addition or subtraction. In the expression \((\sqrt{3} + x)^2\), \(\sqrt{3}\) and \(x\) are the two terms, making it a binomial.
Understanding binomials is crucial, as they form the basis for many algebraic operations, especially when dealing with polynomials.

• They are simple but foundational elements in algebra.
• Binomials often appear in quadratic expressions, calculus, and even complex numbers.

Recognizing binomials becomes second nature with practice and is the first step in managing more complex algebraic expressions.

Square of a Sum Formula

The Square of a Sum Formula is an essential algebraic concept used to expand the square of a binomial. It states that for any two numbers \(a\) and \(b\), the square of their sum is given by:

• \((a+b)^2 = a^2 + 2ab + b^2\)

This formula is crucial because it simplifies the process of expanding binomials.
In our example, with \(a = \sqrt{3}\) and \(b = x\), applying the formula involves computing three parts:

• The square of the first term: \((\sqrt{3})^2 = 3\)
• Twice the product of both terms: \(2 \times \sqrt{3} \times x = 2\sqrt{3}x\)
• The square of the second term: \(x^2\)

Understanding and applying this formula can greatly increase efficiency when dealing with polynomials and is often a favorite topic in algebra for its straightforward nature.

Simplification

Simplification is a key process in algebra where expressions are rewritten in a simpler or more standard form. In the context of our problem, simplification involved combining the results from using the Square of a Sum Formula.
The expression \(3 + 2\sqrt{3}x + x^2\) is considered simplified because:

• All terms are distinct; none can be combined further.
• Simplification often stops when no further operations can make the expression more concise.

This process helps in making expressions easier to read and interpret.
Mastering simplification skills can make it easier to compare expressions, solve equations, and identify properties of algebraic structures.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In the expression \((\sqrt{3} + x)^2\), the exponent \(2\) indicates that the binomial should be multiplied by itself.
Key aspects of exponents include:

• The base is the number being multiplied (\(\sqrt{3} + x\) in this case).
• Exponents denote the number of times the base is used as a factor: \((\sqrt{3} + x) \times (\sqrt{3} + x)\).

Using exponents helps simplify expressions and convey large computations compactly.
Understanding and manipulating exponents is fundamental in algebra and beyond, such as in geometric sequences and growth-related problems.