Problem 27 - Algebra A Combined Function

Question

Multiply. $$ \left(y-\frac{2}{7}\right)^{2} $$

Step-by-Step Solution

Answer

\(y^2 - \frac{4}{7}y + \frac{4}{49}\)

Step 1: Understand the Expression

The given expression is \( \left(y - \frac{2}{7}\right)^2 \). This represents a binomial square, which can be expanded using the formula \((a - b)^2 = a^2 - 2ab + b^2\).

Step 2: Identify Terms ‘a’ and ‘b’

In the expression \( \left(y - \frac{2}{7}\right)^2 \), identify \( a = y \) and \( b = \frac{2}{7} \).

Step 3: Compute \(a^2\)

Calculate \(a^2 = y^2\).

Step 4: Compute \(2ab\)

Calculate \(2ab = 2 \cdot y \cdot \frac{2}{7} = \frac{4}{7}y\).

Step 5: Compute \(b^2\)

Calculate \(b^2 = \left(\frac{2}{7}\right)^2 = \frac{4}{49}\).

Step 6: Write the Expanded Form

Combine the results from Steps 3, 4, and 5 using the formula: \(y^2 - \frac{4}{7}y + \frac{4}{49}\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, these symbols, often represented by letters, are used to represent numbers and quantities in formulas and equations. This area of mathematics is fundamental in solving problems involving unknowns, as it provides a systematic way to express and solve equations.
When working with algebraic expressions like the given binomial \( (y - \frac{2}{7})^{2} \), understanding how to manipulate these symbols is crucial. You will often encounter expressions that can be multiplied, expanded, and simplified, as is the case with this exercise. Grasping the basics of algebra will assist you in recognizing patterns and applying appropriate formulas, such as the square of a binomial, to solve equations efficiently.
Algebra forms the basis for many more complex areas of mathematics and is used in various real-world applications, from engineering and physics to economics and beyond. It's essential for developing logical thinking and problem-solving skills.

Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. These expressions are fundamental in algebra and appear in many different contexts. A classic example of a quadratic expression emerges when expanding a binomial square.
In this exercise, the expression \( (y - \frac{2}{7})^{2} \) is expanded into \( y^2 - \frac{4}{7}y + \frac{4}{49} \), forming a quadratic expression. Here, \( y^2 \) is the quadratic term, \( -\frac{4}{7}y \) is the linear term, and \( \frac{4}{49} \) is the constant term.
Each term plays a specific role; the coefficient of the quadratic term indicates the parabolic nature of its graph, while the linear and constant terms adjust its orientation and position. Quadratic expressions are not only vital for solving quadratic equations but also for understanding the properties of quadratic functions, such as their graphs, roots, and vertex.

Polynomial Multiplication

Polynomial multiplication is a method of multiplying two algebraic expressions to produce another polynomial. This technique involves distributing each term in one polynomial to every term in the other, following the distributive property. Polynomials, like quadratics, can be multiplied by applying formulas such as the binomial square.
In the example given, multiplying \( (y - \frac{2}{7})^{2} \) involves applying the formula \( (a - b)^2 = a^2 - 2ab + b^2 \). This formula allows you to expand the binomial quickly without manually distributing every term.

• First, identify the terms as \( a = y \) and \( b = \frac{2}{7} \).
• Calculate each part separately: \( a^2 = y^2 \), \( 2ab = \frac{4}{7}y \), and \( b^2 = \frac{4}{49} \).
• Combine these to get the expanded polynomial: \( y^2 - \frac{4}{7}y + \frac{4}{49} \).

Understanding polynomial multiplication is essential because it forms the foundation for more complex algebraic manipulations. It also emphasizes the importance of recognizing and applying algebraic formulas to simplify and solve mathematical problems efficiently.