Problem 23 - Elementary Algebra

Question

Find the following products and simplify. $$ (x+6)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified expression for \((x+6)^{2}\) is \(x^2 + 12x + 36\).

Step 1: Rewrite the expression as a product

First, rewrite the square of (x+6) as a product of two binomials: $$ (x+6)^{2} = (x+6)(x+6) $$

Step 2: Apply the distributive property to the product (FOIL)

We distribute each term from the first binomial to each term from the second binomial: $$ (x+6)(x+6) = x\cdot x + x\cdot 6 + 6\cdot x + 6\cdot 6 $$

Step 3: Multiply and combine the terms

Now, we multiply the terms and combine the like terms: $$ x^2 + 6x + 6x + 36 = x^2 + 12x + 36 $$

Step 4: Write the final simplified expression

The simplified expression for \((x+6)^{2}\) is: $$ (x+6)^{2} = x^2 + 12x + 36 $$

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra. They are combinations of numbers, variables, and operations. For instance, the expression \((x + 6)^2\) is an algebraic expression. It shows a variable \(x\) plus a constant 6, all squared. Understanding algebraic expressions is essential because they form the language through which math concepts are communicated. They allow us to represent real-world situations symbolically. To manipulate these expressions, knowing how to apply operations correctly is key. This includes operations such as addition, subtraction, multiplication, and division. Learning to work with algebraic expressions includes understanding terms, coefficients, and the order of operations, which are critical for solving equations and inequalities. When you see expressions, you should recognize how terms connect and can be simplified or expanded within mathematical rules.

Distributive Property

The distributive property is a fundamental algebra rule that allows us to multiply terms inside parentheses by distributing one term over the terms inside. For example, we used it here with \((x + 6)(x + 6)\):

• Distribute \(x\) across the binomial \((x + 6)\).
• Distribute 6 across \((x + 6)\) as well.

This gives: - \( x \cdot x + x \cdot 6 + 6 \cdot x + 6 \cdot 6 \). By applying the distributive property, every term in the first binomial is multiplied by every term in the second binomial. This method is often remembered with the acronym FOIL (First, Outer, Inner, Last) when dealing with two binomials. Using the distributive property effectively simplifies algebraic expressions and allows us to expand squared terms, like \((x + 6)^2\), into a simpler form. It is essential for solving equations that require simplification or expansion of products of expressions.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. Once an algebraic expression is expanded, combining like terms is the key step. For the expanded form \(x^2 + 6x + 6x + 36\), we simplify by adding the like terms:

• Combine the \(6x\) and \(6x\) to get \(12x\).
• The expression simplifies to \(x^2 + 12x + 36\).

Simplifying is crucial as it makes algebraic expressions easier to understand and work with. It helps in solving equations and can reveal patterns or solutions more clearly. Also, a simplified expression is less prone to arithmetic errors. Whether adding, subtracting, or combining alike or different terms, always follow mathematical rules to ensure that expressions maintain their integrity. Simplifying is the final touch to many algebraic operations, bringing clarity and order to mathematical expressions.