Problem 178 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Use the special product \((A+B)^{2}=A^{2}+2 A B+B^{2}\) to multiply: \((\sqrt{x+4}+1)^{2}\).

Step-by-Step Solution

Answer

The result of multiplying \((\sqrt{x+4}+1)^{2}\) using the special product is \(x+2\sqrt{x+4}+5\).

Step 1: Identification of A and B

From the expression \((\sqrt{x+4}+1)^{2}\), we identify \(A\) as \(\sqrt{x+4}\) and \(B\) as \(1\).

Step 2: Expanding expression using the special product

Now substitute \(A\) and \(B\) into the special product formula \((A+B)^{2}=A^{2}+2 A B+B^{2}\). This gives us \((\sqrt{x+4}+1)^{2}=(\sqrt{x+4})^{2}+2 \cdot \sqrt{x+4} \cdot 1+1^{2}\).

Step 3: Solving further

Solving further, we get \(x+4 + 2\sqrt{x+4} + 1\). Reordering, we obtain \(x + 2\sqrt{x+4}+5\).

Key Concepts

Binomial Expansion

Binomial expansion is a process used in algebra to expand expressions that involve two terms, known as binomials, raised to a power. The term "binomial" refers to the expression having two parts: a component A and a component B. When we refer to binomial expansion, we specifically look at expanding binomials raised to an integer power, such as \((A + B)^n\).

In this context, a commonly used binomial expansion is the square of a binomial, represented by the formula:

• \((A + B)^2 = A^2 + 2AB + B^2\)

This formula shows us how to expand a binomial squared into a sum containing three terms. Each term results from multiplying the components A and B in different combinations and squaring them.

Binomial expansion is helpful in simplifying complex expressions and solving algebraic equations involving binomials. Understanding how to use this formula helps achieve accuracy in calculations and faster solutions.

Square of a Binomial

The square of a binomial is a specific case of binomial expansion where the exponent is 2. This means we are focusing on expressions of the form \((A + B)^2\). The formula \((A + B)^2 = A^2 + 2AB + B^2\) helps simplify the multiplication process involved when a binomial is squared.

In this case, here is what happens:

• First Term: Square of the first term (\(A^2\))
• Middle Term: Twice the product of both terms (\(2AB\))
• Last Term: Square of the last term (\(B^2\))

Let's look at the example: \((\sqrt{x+4}+1)^2\). By identifying \(A\) as \(\sqrt{x+4}\) and \(B\) as \(1\), we apply the formula to obtain an expanded expression. This results in an expression that is more comfortable to work with for further calculations.

Understanding the square of a binomial eliminates a lot of manual expansion work, making it a quick and reliable method to simplify expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(x\)), and operations (such as addition, subtraction, multiplication, etc.). They are the basic building blocks of algebra and used in mathematical modeling of real-world phenomena.

Key components of algebraic expressions include:

• Variables: Symbols that represent unknown values or quantities.
• Constants: Fixed values that do not change.
• Operators: Symbols that denote mathematical operations.

Consider the expression \((\sqrt{x+4} + 1)^2\), which involves simplifying it using algebraic rules. The solution involves identifying the parts of the expression and using them effectively in formulas like the square of a binomial.

When dealing with algebraic expressions, it’s essential to understand how to manipulate variables and constants. This manipulation often requires employing special techniques, such as those from binomial expansion and recognizing common patterns like squares, leading to more straightforward and efficient problem-solving methods.