Problem 25 - Algebra 2

Question

Write a verbal expression to represent each equation. \(y^{2}=4 y\)

Step-by-Step Solution

Answer

y squared equals four times y.

Step 1: Identifying equation components

The equation given is \(y^2 = 4y\). Let's break it down into two parts: \(y^2\) is 'y squared' and \(4y\) is 'four times y'.

Step 2: Constructing the verbal expression

Now that we have identified the components, we can write the equation as a verbal expression. Since \(y^2\) appears on the left side and is equal to \(4y\), we say 'y squared equals four times y.'

Key Concepts

Equations

An equation is a mathematical statement that asserts the equality of two expressions. It consists of two sides separated by an equal sign (=). Each side of the equation represents a mathematical expression, and the basic purpose is to find the values that make the equation true. In the context of the original exercise, the equation is given as \(y^2 = 4y\). Here,

• \(y^2\) is the expression on the left side.
• \(4y\) is the expression on the right side.

The process of problem solving in equations usually involves manipulating these expressions to uncover the unknown values or to represent the equation in a different form. Equations are foundational tools in mathematics used to describe relationships, solve problems, and model real-world situations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators, such as addition, subtraction, multiplication, and division. These expressions form the building blocks of algebraic equations. In the equation \(y^2 = 4y\), both \(y^2\) and \(4y\) are algebraic expressions.

These expressions:

• Can contain constants (like 4 in the expression \(4y\)).
• Include variables (such as \(y\) in both expressions).
• Use operators, like multiplication indicated implicitly in \(4y\) as "four times \(y\)".

Understanding how to manipulate and interpret algebraic expressions is crucial in solving equations. Expressions are often transformed or simplified by applying mathematical operations to isolate variables or to express them in a way that's easier to work with.

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest degree of the variable is squared. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). In the original exercise, the equation is \(y^2 = 4y\), which can be rearranged to fit the standard form of a quadratic equation:

• By subtracting \(4y\) from both sides, it transforms to \(y^2 - 4y = 0\).
• This makes it evident as a quadratic since there's an \(y^2\) term and the equation equals zero.

Quadratics often require specific methods for solving, such as factoring, the quadratic formula, or completing the square. Quadratic equations appear frequently in various scenarios, from physics problems involving projectile motion to economics for cost functions. Understanding their structure and solution methods is vital in higher-level algebra.