Problem 10 - Elementary Algebra

Question

Determine the values of \(a, b,\) and \(c\) in the following quadratic equations. $$ y^{2}=0 $$

Step-by-Step Solution

Answer

Answer: In the quadratic equation \(y^2 = 0\), the values of a, b, and c are: a = 1 b = 0 c = 0

Step 1: Identify the equation in standard form

We are given the equation: $$ y^2 = 0 $$ To write this in the standard form of a quadratic equation, we write it as: $$ 1y^2 + 0y + 0 = 0 $$ Now we can easily identify the values of a, b, and c.

Step 2: Determining the values of a, b, and c

Comparing the standard form equation to the defined form of a quadratic equation (\(ax^2 + bx + c = 0\)), we find the values of a, b, and c as follows: a = 1 (coefficient of \(y^2\)) b = 0 (coefficient of y) c = 0 (constant term)

Step 3: Conclusion

In the given quadratic equation \(y^2 = 0\), the values of a, b, and c are: a = 1 b = 0 c = 0

Key Concepts

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is an essential concept often used in algebra. It helps us to easily organize, identify, and analyze the different parts of these equations. A quadratic equation in its standard form is written as:
\[ ax^2 + bx + c = 0 \]
Here, the variables 'a', 'b', and 'c' are referred to as coefficients, and 'x' represents the variable of the equation. The coefficient 'a' must be non-zero to ensure the equation is indeed quadratic, as the presence of the \(x^2\) term defines the equation's quadratic nature. It's this structure that allows us to systematically approach solving quadratic equations, whether by factoring, completing the square, or using the quadratic formula. Understanding this form is critical in identifying the coefficients, which leads us to the next concept.

Identifying Coefficients

In the context of quadratic equations, identifying coefficients is straightforward once you recognize the standard form. The coefficients 'a', 'b', and 'c' are the numerical factors of the terms \(x^2\), 'x', and the constant term, respectively. In the exercise given:
\[ y^2 = 0 \]
we convert this to the familiar standard form to identify the coefficients:
\[ 1y^2 + 0y + 0 = 0 \]
Here, the coefficient of \(y^2\) is 1 (even though it’s not explicitly written, any variable without a coefficient has an implicit coefficient of 1). The coefficient of 'y' and the constant term are both 0. This identification process is key for algebraic problem solving as it sets the stage for finding solutions to the equation.

Algebraic Problem Solving

Algebraic problem solving involves a series of steps and methods used to find the solutions to various types of algebraic equations, including quadratic equations. In our exercise, we used the basic principle of bringing the equation to its standard form to facilitate the identification of coefficients. This methodical approach prepares us for the problem-solving process, which may involve factoring, applying the quadratic formula, graphing, or other algebraic methods. Regardless of the technique used, the problem-solving process typically includes setting up the equation, identifying coefficients, choosing an appropriate method, and then applying that strategy to find the value of the variable. The goal is to isolate the variable and solve for its values that satisfy the equation. In this way, a systematic approach simplifies the problem and makes it easier to understand and solve.