Problem 39 - Elementary Algebra

Question

For the following problems, find the products. $$ \left(y-\frac{5}{6}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Expand the expression \((y-\frac{5}{6})^2\). Answer: \(y^2 - \frac{10}{6}y + \frac{25}{36}\).

Step 1: Identify a and b

In the given expression, let \(a = y\) and \(b = \frac{5}{6}\).

Step 2: Apply the formula

Now, use the formula \((a-b)^2 = a^2 - 2ab + b^2\) to expand the expression. Substitute \(a\) and \(b\) with their respective values: \((y-\frac{5}{6})^2 = y^2 - 2(y)(\frac{5}{6}) + (\frac{5}{6})^2\)

Step 3: Calculate the terms

We will calculate each term of the expanded expression: \(y^2\) remains the same. The term \(2(y)(\frac{5}{6}) = \frac{10}{6}y\). The term \((\frac{5}{6})^2 = \frac{25}{36}\).

Step 4: Combine the terms

Combine all the calculated terms: \((y-\frac{5}{6})^2 = y^2 - \frac{10}{6}y + \frac{25}{36}\)

Step 5: Simplify the expression (if possible)

The expression is already in its simplest form. Therefore, \((y-\frac{5}{6})^2 = y^2 - \frac{10}{6}y + \frac{25}{36}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like y), and operation symbols. They are essential parts of algebra and enable us to represent relationships and formulas in a concise way. Think of these expressions as sentences in the language of mathematics.

For instance, \( (y - \frac{5}{6})^2 \) is an algebraic expression. It includes the variable \(y\), a numerical fraction \(\frac{5}{6}\), and the exponent \(2\) which indicates the expression is squared. When dealing with algebraic expressions, it's important to be familiar with the terms used, such as:

• Coefficient: The numerical factor of a term that contains a variable. For example, in \(\frac{10}{6}y\), \(\frac{10}{6}\) is the coefficient.
• Variable: A symbol that represents an unknown number. In our expression, \(y\) is the variable.
• Constant: A value that doesn’t change. In the term \(\frac{25}{36}\), this is a constant because it's a fixed number.

Understanding how to construct and interpret algebraic expressions is fundamental to mastering higher-level math problems. It sets the stage for solving equations and simplifying complex mathematical concepts.

Square of a Binomial

The square of a binomial is a specific type of algebraic expression. A binomial is simply a polynomial with two terms. For example, \(a + b\) and \(a - b\) are both binomials. The square of a binomial refers to multiplying the binomial by itself, which is denoted as \( (a + b)^2 \) or \( (a - b)^2 \).

There is a classic algebraic formula to find the square of a binomial: the \(a^2 + 2ab + b^2\) pattern for \( (a + b)^2 \) and the \(a^2 - 2ab + b^2\) pattern for \( (a - b)^2 \). When you apply this formula, it eliminates the need for lengthy multiplication and can significantly simplify your calculations.

Let's consider our original expression \( (y - \frac{5}{6})^2 \). By recognizing this expression as the square of a binomial, you can use the corresponding formula to expand and simplify it in just a few steps:

• First term: \(y^2\), which is \(a^2\).
• Second term: \( -2(y)(\frac{5}{6}) \) or \( -\frac{10}{6}y \) following the pattern \( -2ab \).
• Third term: \( (\frac{5}{6})^2 \) which is \(b^2\), equal to \(\frac{25}{36}\).

Knowing this pattern helps in quickly squaring binomials without needing to perform the multiplication manually.

Simplifying Expressions

Simplifying expressions is a process of making an algebraic expression as easy as possible to understand or solve by reducing it to its most basic form. This involves combining like terms, reducing fractions, and removing parentheses when possible.

In our step-by-step solution, simplification happens in the final step after expanding the square of the binomial. Each term has been calculated, and now they have to be combined. Fortunately, our example already results in its simplest form after combining the terms, so no further simplification is needed.

Simplification can often involve:

• Combining like terms, which are terms that have identical variable parts.
• Reducing fractions to their lowest terms by finding the greatest common divisor for the numerator and the denominator.
• Applying various algebraic identities to make expressions easier to work with.

Sometimes, simplification might also involve factoring expressions or applying the distributive property in reverse. In essence, simplifying is about making your expression clean and less cluttered, without changing its value or meaning and is a critical step in solving algebraic problems.