Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 4 x^{2}-24 x+36=0

Step-by-Step Solution

Answer

The equation represents a degenerate conic, specifically a single point at \(x = 3\).

Step 1: Recognize the Form of the Equation

The given equation is \(4x^2 - 24x + 36 = 0\). Notice that this is a quadratic equation in the form \(ax^2 + bx + c = 0\). Our task is to determine if it can represent a conic section.

Step 2: Simplify the Equation

Let's start by dividing the entire equation by 4 to make it easier to complete the square. This gives us \(x^2 - 6x + 9 = 0\).

Step 3: Complete the Square

Take the \(x^2 - 6x\) part and complete the square. To do this, take half of the coefficient of \(x\), which is -6, divide by 2 to get -3, then square it to get 9.

Step 4: Rewrite the Equation

The equation \(x^2 - 6x + 9 = 0\) can now be rewritten as \((x-3)^2 = 0\) because \((x-3)^2 = x^2 - 6x + 9\).

Step 5: Analyze for Conic Section

The equation \((x-3)^2 = 0\) is a degenerate case of a conic section. It represents a single point rather than an ellipse, hyperbola, or parabola.

Key Concepts

Quadratic Equations

Quadratic equations are foundational in understanding conic sections. A quadratic equation is typically expressed in the standard form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, \(a eq 0\). Every quadratic equation graphs as a parabola, and its shape can be seen as part of a conic section depending on its coefficients and terms. - If there are only \(x^2\) terms, the graph may be a parabola.- Combining \(x^2\), \(y^2\), and even other linear terms can create ellipses, hyperbolas, or circles based on the condition of the equation.In the case provided: 4x^2 - 24x + 36 = 0,since there is only one variable \(x\) present, our focus is on its quadratic form. Later steps would simplify this equation to better reveal its characteristics.

Completing the Square

Completing the square is a useful technique used for rewriting quadratic equations in a form that easily shows their roots and transformations. It involves rearranging the terms of a quadratic equation to create a perfect square trinomial. Here's how we apply this method to the quadratic equation: 1. Start with the equation: \( x^2 - 6x + 9 = 0 \) 2. Focus on the quadratic and linear terms, \(x^2 - 6x\). The mission is to transform this into a perfect square. 3. To make a perfect square, take half of the coefficient of \(x\), then square it. For \(-6\), half is \(-3\) and \((-3)^2\) equals \(9\).4. The expression can then be rewritten as: \((x-3)^2\) because \((x-3)^2 = x^2 - 6x + 9\). This type of transformation simplifies solving equations and analyzing potential conic sections.

Degenerate Conic

The term 'degenerate conic' refers to a special case in which the usual properties of conic sections converge to represent something unusual, like a point or a line, rather than the more familiar shapes such as parabolas, ellipses, or hyperbolas.The completed expression \((x-3)^2 = 0\),leads to the realization that there's not a curve of interest, but instead a singular solution or point. Essentially, this equation resolves to\(x - 3 = 0\), or \(x = 3\).In the context of conic sections, when the elements collapse to a point, it shows that this is a 'degenerate' outcome. It helps one understand edge cases in conic section studies, reflecting the wide variety of forms and intersections these equations can take.