Problem 55 - Algebra 2

Question

Each equation is of the form \(A x^{2}+B x y+C y^{2}+D x+\) \(E y+F=0 .\) Identify the values of \(A, B,\) and \(C\). $$ x^{2}-4 x+4 y+2=0 $$

Step-by-Step Solution

Answer

A = 1, B = 0, C = 0

Step 1: Identify the general quadratic form

The general form for a quadratic equation in two variables is given by \( Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 \). We need to match this form with the given equation \( x^{2} - 4x + 4y + 2 = 0 \) to find the values of the coefficients \( A, B, \) and \( C \).

Step 2: Identify coefficient A

In the equation \( x^{2} - 4x + 4y + 2 = 0 \), the term \( x^2 \) appears with the coefficient 1. Thus, \( A = 1 \) because it matches with \( Ax^2 \) in the general form.

Step 3: Identify coefficient B

The equation does not contain a term of the form \( xy \). In other words, \( xy \) is absent, which implies \( B = 0 \).

Step 4: Identify coefficient C

The equation \( x^{2} - 4x + 4y + 2 = 0 \) does not have a \( y^{2} \) term. Therefore, there is no corresponding coefficient for \( y^2 \), which means \( C = 0 \).

Key Concepts

General Form of Quadratic Equation

The general form of a quadratic equation in two variables is a versatile format used to express a wide range of quadratic relations. Understanding this form is key to analyzing and manipulating these equations. It is expressed as:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]where each term represents different aspects of the quadratic equation:

• \(Ax^2\): Represents the quadratic term involving only \(x\).
• \(Bxy\): Represents the interaction between both variables \(x\) and \(y\).
• \(Cy^2\): Represents the quadratic term involving only \(y\).
• \(Dx, Ey, F\): Represent the linear terms and constant part.

This form simplifies the process of identifying different coefficients for classes of quadratic equations, aiding in their classification and solution.

Identifying Coefficients

Identifying coefficients in a quadratic equation helps us understand the structure and characteristics of the equation. Let's analyze the given equation:\[ x^2 - 4x + 4y + 2 = 0 \]In the context of the general form, we need to pinpoint what each coefficient stands for:

• Coefficient \(A\): Corresponds to the \(x^2\) term. Here, \(A = 1\).
• Coefficient \(B\): Corresponds to the \(xy\) term, which is absent. Thus, \(B = 0\).
• Coefficient \(C\): Corresponds to the \(y^2\) term, also absent. Hence, \(C = 0\).

Identifying these coefficients correctly allows us to further analyze the nature of the quadratic equation and forms the basis for more advanced solutions.

Two-variable Quadratic Equation

A two-variable quadratic equation extends the concept of quadratic equations to include relationships between two variables. These equations are often used in geometry, physics, and other fields requiring interactions between two entities. Such equations might appear in several contexts, such as:

• Defining a conic section (ellipse, parabola, hyperbola) based on the values of \(A, B,\) and \(C\).
• Modeling scenarios where two factors have quadratic relations to each other.

The general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) allows these equations to be widely applicable and adaptable to various mathematical and real-world problems. Each coefficient reflects aspects of both individual and mutual interactions between \(x\) and \(y\), making it a core concept for understanding multi-variable equations.