Question

Show that $$n^{2}+n+2(n+1)=(n+1)^{2}+(n+1)$$

Short answer

Both sides simplify to \(n^2 + 3n + 2\), confirming the equation is true.

Step-by-step solution

Reorganize the Original Equation

Start by taking the original equation: \(n^2 + n + 2(n+1) = (n+1)^2 + (n+1)\). Notice the structure on both sides and how the equation is organized.

Expand Expressions

Expand each term in the equation. The left side becomes: \[n^2 + n + 2(n+1) = n^2 + n + 2n + 2.\]The right side expands to: \[(n+1)^2 + (n+1) = (n^2 + 2n + 1) + (n + 1).\]

Simplify Both Sides

Now, simplify the expressions on both sides.For the left side: \[n^2 + n + 2n + 2 = n^2 + 3n + 2.\]For the right side: \[n^2 + 2n + 1 + n + 1 = n^2 + 3n + 2.\]

Compare the Simplified Forms

Compare the simplified forms from both sides of the equation.Left: \(n^2 + 3n + 2\)Right: \(n^2 + 3n + 2\)Since the expressions on both sides are exactly equal, the original equation is true for all values of \(n\).

Key concepts

Algebraic Equations

Algebraic equations are like puzzles made from numbers, letters, and operations. They form statements that show the equality between two expressions. In this exercise, the equation we are dealing with is \(n^2 + n + 2(n+1) = (n+1)^2 + (n+1)\). Here, our job is to verify if this specific mathematical relationship holds true for any integer \(n\). Remember, algebraic equations often need simplifying or expanding to truly understand and prove their truth.

Understanding these basic elements of algebraic equations is crucial:

• The variables, like \(n\) in this case, represent unknown numbers that we aim to solve or prove in relation.
• Constants, such as the numbers 1, 2, and elsewhere, interact with these variables according to given operations.
• The equality sign signals that both expressions on either side must be equivalent after all calculations are complete.

Algebraic equations demand both an analytical mind and careful calculation as they require thorough manipulation and analysis of their components to verify or solve them successfully.

Simplification

Simplification is a key skill in solving algebraic equations. It involves reducing expressions to their simplest and most concise form. This process uncovers the core elements of the expression, making it easier to work with and understand. During simplification, we combine like terms and perform arithmetic operations.

In the given exercise, after expanding the expressions, we simplified both sides by combining like terms:

• The left side \(n^2 + n + 2n + 2\) is simplified to \(n^2 + 3n + 2\).
• The right side \(n^2 + 2n + 1 + n + 1\) also simplifies to \(n^2 + 3n + 2\).

This shows how despite the appearance of complexity, simplification reveals true equality by breaking down expressions into more manageable forms. Consistent practice in simplification helps in recognizing equivalent terms and errors in calculations. With every simplification step, clarity and accuracy increase, leading to correct conclusions about the original equation.

Expanding Expressions

Expanding expressions is the process of removing grouping symbols, such as parentheses, to spread out an expression into its fullest form. This technique often utilizes the distributive property to open brackets and multiply each term inside a set of parentheses by the term outside of it.

For example, in the exercise:

• The left side \(n^2 + n + 2(n+1)\) was expanded to \(n^2 + n + 2n + 2\).
• The right side \((n+1)^2 + (n+1)\) expanded to \((n^2 + 2n + 1) + (n + 1)\).

Through expansion, the hidden operations within groups are revealed, allowing for effective combination and simplification. It gives a clearer view of all the terms involved and how they interact. Mastering expansion helps in understanding the structure of more complex equations and is essential for ensuring parts of an equation are simplified accurately.