Problem 43 - Elementary Algebra

Question

For the following problems, find the products. $$ (a+0.5)^{2} $$

Step-by-Step Solution

Answer

Answer: The product of the given expression is a^2 + a + 0.25.

Step 1: Identify the binomial

In this case, the binomial is \((a + 0.5)\).

Step 2: Apply the formula for the square of a binomial

The formula for the square of a binomial \((a+b)^2\) is given by \(a^2 + 2ab + b^2\). Here \(a\) represents the first term of the binomial and \(b\) represents the second term of the binomial.

Step 3: Substitute the values of \(a\) and \(b\) into the formula

In our case, \(a\) is represented by the variable \(a\) and \(b\) is equal to \(0.5\). Therefore, substitute these values into the formula: \((a+0.5)^{2} = a^2 + 2(a)(0.5) + (0.5)^{2}\)

Step 4: Simplify the expression

Now, simplify the expression by performing the operations: \(a^2 + 1a + 0.25 = a^2 + a + 0.25\) So, the product of the given expression \((a+0.5)^{2}\) is equal to \(a^{2} + a + 0.25\).

Key Concepts

Square of a Binomial

The square of a binomial is a fundamental concept in algebra, often encountered when dealing with expressions that involve two terms separated by a plus or minus sign, known as binomials. Squaring a binomial means multiplying the binomial by itself. For example, in the exercise \((a + 0.5)^2\), we are asked to find the square of the binomial \((a + 0.5)\). The standard formula for the square of a binomial is given by \((a + b)^2 = a^2 + 2ab + b^2\).
This formula provides a way to expand the expression without having to multiply \((a + 0.5)\) by itself through lengthy distribution processes. Instead, we can directly substitute \(a\) and \(b\) into the formula, where \(a\) is the first term and \(b\) is the second term. In the context of our example, \(b\) equals \(0.5\).
Using the formula helps streamline calculations and allows us to quickly determine the expanded form, as shown in the solution:

• \(a^2\) is the square of the first term.
• \(2ab\) gives us the twice the product of the two terms.
• \(b^2\) is the square of the second term.

This approach not only simplifies squaring but also enhances our understanding of how algebraic expressions behave.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together form a mathematical statement. In the problem at hand, our primary expression is \((a + 0.5)^2\), an example of an algebraic expression that needs to be simplified.
Components of algebraic expressions include terms, coefficients, and constants. For instance, in the expression \(a^2 + a + 0.25\), we have three terms. The \(a^2\) term includes a variable and its exponent.
The term \(a\) is simple and represents a coefficient multiplied by a variable, where the coefficient is \(1\) (often omitted because multiplying by one does not change the term).
The \(0.25\) is a constant term, having no variable tied to it. Understanding algebraic expressions requires recognizing these parts and knowing how to manipulate them, such as combining like terms or using formulas for simplification. The structure of these expressions allows for various algebraic operations, which are core skills in mathematics and science fields. Knowing how to expand, simplify, or factor these expressions enables better comprehension of more complex problems.

Polynomial Multiplication

Polynomial multiplication is a process where two or more polynomials are multiplied together. This concept applies to our exercise because we started with a binomial \((a + 0.5)\) that needed to be squared. Squaring a binomial is a special case of polynomial multiplication.
Generally, when multiplying polynomials, each term in the first polynomial is multiplied by every term in the second. For example, if we directly multiply \((a + 0.5)(a + 0.5)\), it involves multiplying \(a\) by \(a\) and \(0.5\), then \(0.5\) by \(a\) and \(0.5\), resulting in four products to add up:

• \(a \cdot a = a^2\)
• \(a \cdot 0.5 = 0.5a\)
• \(0.5 \cdot a = 0.5a\)
• \(0.5 \cdot 0.5 = 0.25\)

Simplifying these gives \(a^2 + a + 0.25\), as both \(0.5a\) terms add up to \(a\).
Using the formula for the square of a binomial simplifies this process, leading directly to the result without having to calculate each product separately. Understanding polynomial multiplication builds a foundation for many other algebraic concepts, providing a stepping stone to more advanced topics.