Problem 103 - Beginning and Intermediate Algebra

Question

Find the following special products. $$\quad(2 d-5)^{2}$$

Step-by-Step Solution

Answer

The short answer to the question is: \((2d - 5)^2 = 4d^2 - 20d + 25\).

Step 1: Identify the binomial square rule

Recall the binomial square formula: \((a - b)^2 = a^2 - 2ab + b^2\). Here, we have the expression \((2d - 5)^2\), so we can identify \(a\) and \(b\): \(a = 2d\) \(b = 5\)

Step 2: Substitute values of a and b into the formula

Substitute the values of \(a = 2d\) and \(b = 5\) into the binomial square formula: \((2d - 5)^2 = (2d)^2 - 2(2d)(5) + (5)^2\)

Step 3: Simplify the expression

Now, simplify the expression by evaluating the powers and performing the multiplications: \((2d - 5)^2 = (4d^2) - 20d + 25\) So, the special product of \((2d - 5)^2\) is \(4d^2 - 20d + 25\).

Key Concepts

Binomial Square Rule

The binomial square rule is a special product formula used in algebra to simplify expressions involving the square of a binomial. A binomial is an algebraic expression that contains two terms, such as \((a - b)\). When we square such an expression, we apply the formula: \((a - b)^2 = a^2 - 2ab + b^2\). This formula helps to turn the operation of squaring a binomial into a simpler algebraic expression without having to perform lengthy multiplications. It accounts for both the squares of the individual binomial terms as well as the product of these terms. Using this rule reduces time and effort as it eliminates repetitive calculation steps, especially when dealing with complex numbers or multiple algebraic terms. If you remember this rule, you can quickly expand binomial squares in your algebra homework.

Algebraic Expressions

An algebraic expression is a mathematical statement using numbers, variables, and operational symbols. In algebra, these expressions serve as the foundation for solving equations and simplifying complex mathematical problems. Common components of an algebraic expression include:

• **Variables**, such as \(x\), \(y\), or \(d\)
• **Constants**, which are fixed numbers like 5 or 10
• **Coefficients**, which multiply variables, e.g., 4 in \(4d\)
• **Operators**, including addition (+), subtraction (-), multiplication (×), and division (÷)

In the exercise, the expression \((2d - 5)^2\) is made up of two terms, with \(2d\) as the first term and 5 as the constant second term. Understanding each part's role in the expression helps when applying rules like the binomial square, thereby making simplification possible.

Simplification in Algebra

Simplification in algebra is the process of making an expression easier to understand and work with by combining like terms and reducing expressions. Simplifying expressions is key to solving problems efficiently and accurately. To simplify the expression \((2d - 5)^2\) using the binomial square rule, you follow these steps:

• Identify each part of the expression, noting \(a = 2d\) and \(b = 5\).
• Apply the binomial square rule: Substitute the identified values in the formula \((a - b)^2 = a^2 - 2ab + b^2\).
• Evaluate the individual squares and multiply appropriately: \((2d - 5)^2 = (4d^2) - 20d + 25\).
• Combine like terms and ensure no further simplification is possible.

By simplifying expressions, we make a mathematical statement clearer, allowing a better understanding of relationships between different components and ultimately solving equations more straightforwardly.