Problem 5 - Algebra and Trigonometry

Question

The Special Product Formula for the “square of a sum” is \((A+B)^{2}=\) ________. So \((2 x+3)^{2}=\) ______.

Step-by-Step Solution

Answer

\((A+B)^2 = A^2 + 2AB + B^2\); \((2x + 3)^2 = 4x^2 + 12x + 9\).

Step 1: Identify the Special Product Formula

The special product formula for the square of a sum is \((A+B)^2 = A^2 + 2AB + B^2\). We use this formula to expand the square of any binomial sum.

Step 2: Identify A and B

From the expression \((2x + 3)^2\), we can identify \(A = 2x\) and \(B = 3\).

Step 3: Apply the Special Product Formula

Substitute \(A = 2x\) and \(B = 3\) into the formula \((A+B)^2 = A^2 + 2AB + B^2\).

Step 4: Calculate each term in the expansion

Compute each part of the formula: \(A^2 = (2x)^2 = 4x^2\); \(2AB = 2 \cdot 2x \cdot 3 = 12x\); \(B^2 = 3^2 = 9\).

Step 5: Write the expanded expression

Combine the terms to write the expanded form of the expression: \(4x^2 + 12x + 9\).

Key Concepts

Square of a Sum

The square of a sum, often seen in algebra, is a special product formula that helps simplify the expression when you square a binomial. A binomial is any algebraic expression containing two terms, like

• \((A+B)\)
• \((x+y)\).

The formula for the square of a sum, \((A+B)^{2}\), is defined as:\[(A+B)^2 = A^2 + 2AB + B^2\]This formula is very useful because it allows us to expand a binomial square without directly multiplying it by itself. Instead, we:

• Square the first term \(A^2\)
• Add twice the product of the two terms \(2AB\)
• Finally add the square of the second term \(B^2\).

This process not only simplifies calculations but also makes it easier to manage larger expressions. Paying attention to how each part of the formula works is key to mastering more advanced algebraic concepts.

Binomial Expansion

When we talk about binomial expansion, we are referring to the process of using algebraic formulas to expand expressions raised to a power. For example, if we have the expression \((A+B)^2\), its expansion process is governed by the square of a sum formula, which we used in the earlier section:\[(A+B)^2 = A^2 + 2AB + B^2\]In essence, binomial expansion involves finding ways to write a binomial (a two-term expression) in an extended form. Each term of the original binomial, \(A\) and \(B\), will contribute to the final expanded expression:

• The square of the first term \(A^2\)
• The middle term is twice the product of both terms \(2AB\)
• The square of the second term \(B^2\).

Using this structured formula reduces error, especially computation errors, ensuring accuracy when expanding binomials. Binomial expansion is an essential skill in algebra, helping to simplify complex expressions and solve equations.

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics where numbers, variables, and operations come together. These may include:

• Constants like numbers: 3, 5, 10, etc.
• Variables represented by letters like \(x\), \(y\), or \(z\)
• Operations such as addition \((+)\), subtraction \((-))\), multiplication \(\times\), and division \((\div)\).

For instance, the expression \((2x + 3)\) combines a variable and a constant with addition. In algebraic manipulations, such as in special product formulas, understanding how to work with these expressions is crucial.Algebraic expressions can represent relationships and patterns, which is important for modeling real-world situations in areas like physics, finance, and engineering. They are not only used to convey operations between numbers but also serve as a tool for finding unknown values and solving equations. Mastering algebraic expressions provides a strong foundation for tackling more complex mathematical problems.