Problem 1 - Precalculus with Limits

Question

The procedure used to eliminate the \(xy\)-term in a general second-degree equation is called ________ of ________.

Step-by-Step Solution

Answer

The process is known as the Rotation of Axes.

Step 1: Identify the Trigonometric Problem

We examine the given expression: \(xy\).

Step 2: Apply Trigonometric Identities

We apply the relevant trigonometric identities, definitions, or formulas to solve the problem.

Step 3: Simplify and Compute

We simplify the expression using known values and identities.

Step 4: State the Result

The process is known as the Rotation of Axes.

Key Concepts

Completing the Square

To understand the process of completing the square, let's start by considering a quadratic equation of the form ax^2 + bx + c = 0. The goal of completing the square is to transform this equation into a perfect square trinomial plus a constant, which allows for easier solving, particularly for equations where the linear xy-term is present.

The process involves dividing the coefficient of the linear term, b, by 2 and squaring the result to find the number that, when added and subtracted to the equation, converts the quadratic and linear terms into a perfect square. This perfect square will have the form (x + d)^2, where d is the number you obtained. The subtraction of the squared number is done outside the square to maintain the equation's balance.

Example of Completing the Square

For the quadratic equation x^2 + 6x + 5 = 0, we complete the square by adding and subtracting (6/2)^2 = 9 within the equation:

• First, rewrite the equation as x^2 + 6x + 9 - 9 + 5 = 0.
• Next, notice that x^2 + 6x + 9 forms the perfect square (x + 3)^2.
• So, we have (x + 3)^2 - 4 = 0, which can then be solved for x.

Through this method, we haven't eliminated the xy-term, but we've simplified the equation in a way that can be applied to more complex situations, including those with an xy-term.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x, with a general form of ax^2 + bx + c = 0, where ax^2 is the quadratic term, bx is the linear term, and c is the constant term. The coefficients a, b, and c are real numbers, and a cannot be zero.

These equations are fundamental in algebra and have two solutions, which may be real or complex, and are found using a variety of methods including factoring, using the quadratic formula, graphing, or completing the square. One of the biggest challenges when dealing with quadratic equations is the presence of the xy-term, especially when it comes to equations representing conic sections like ellipses or hyperbolas.

Eliminating the xy-term is crucial for simplifying the equation and is typically achieved through rotation of the coordinate system, which does not affect the shape of the graph, but changes its equation to a more manageable form without the xy-term.

Algebraic Methods

The term algebraic methods refers to a variety of techniques used for solving equations and simplifying algebraic expressions. One such method, crucial for handling second-degree equations, is the rotation of axes. This approach is particularly useful for eliminating the xy-term in equations of conic sections, like the equation ax^2 + by^2 + cxy = d.

The process of rotating axes involves changing variables to eliminate the xy-term, consequently simplifying the equation. This is typically done by replacing x and y with new variables x' and y' that are functions of the original variables and the angle of rotation, θ. The rotation effectively removes the xy-term without changing the graph's properties, easing the task of graphing and solving the equation.

Transforming Through Rotation

For example, an equation x^2 - xy + y^2 = 3 can have its axes rotated to remove the xy-term:

• We might use the substitution x = x'cosθ - y'sinθ and y = x'sinθ + y'cosθ.
• Choosing the angle θ such that the xy-term is eliminated after substitution.
• This results in a new equation in x' and y' without the problematic xy-term.

Through these algebraic manipulations, the structure of the equation becomes more straightforward, allowing for easier analysis and resolution.