Problem 53 - 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 \(C\). $$ 2 x^{2}+3 x y-5 y^{2}=0 $$

Step-by-Step Solution

Answer

The values are \(A = 2\), \(B = 3\), and \(C = -5\).

Step 1: Write the General Form

Identify the general form of the equation which is given by \(A x^2 + B x y + C y^2 + D x + E y + F = 0\).

Step 2: Analyze the Given Equation

The given equation is \(2x^2 + 3xy - 5y^2 = 0\). It matches the general form, but lacks some terms. Specifically, the terms involving \(x\), \(y\), and the constant term are absent.

Step 3: Match the Coefficients

Identify corresponding coefficients from the given equation: \(A = 2\) from the term \(2x^2\), \(B = 3\) from the term \(3xy\), and \(C = -5\) from the term \(-5y^2\). The coefficients \(D\), \(E\), and \(F\) are 0 as those terms are absent.

Key Concepts

Coefficients Identification

In quadratic equations, identifying the coefficients is crucial. Coefficients are the numerical factors that multiply the variables in each term. In the equation given,

• The term associated with \(x^2\) gives us the coefficient \(A\).
• The term associated with \(xy\) gives us the coefficient \(B\).
• The term associated with \(y^2\) gives us the coefficient \(C\).

When terms are missing, like a linear \(x\) or \(y\) term or a constant, their coefficients are simply zero. Identifying these coefficients helps understand the direction and shape of the graph of the equation. It also helps determine the nature of its solutions, like whether they are imaginary or real.

General Form of Quadratic Equation

The general form of a quadratic equation in two variables usually looks like: \ \[A x^2 + B xy + C y^2 + D x + E y + F = 0\] \ This representation is universal. It makes identifying relationships between variables easier and sets the stage for analyzing equations.

• \(A, B, C, D, E,\) and \(F\) are the coefficients.
• \(x\) and \(y\) represent the variables.

This form is particularly helpful in identifying the type of conic sections it represents based on the coefficients. The different combinations and values of \(A, B,\) and \(C\) will tell us whether the quadratic is a parabola, ellipse, or hyperbola.

Analyzing Equations

Analyzing quadratic equations involves examining each component to understand its role in the overall equation. Here's what to look for:

• Start by writing the equation in its complete general form, including missing terms with coefficients set to zero.
• Compare the given equation directly against the general form.
• Extract coefficients and determine which terms are omitted.

This practice serves multiple purposes. It clarifies the functionality of each term. It determines which direction the parabola opens, and helps understand the symmetry and shifts in position on a graph. Additionally, it highlights the importance of every term, aiding in predicting real-world applications and behaviors modeled by the equation.