Problem 14 - 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+35=0

Step-by-Step Solution

Answer

The equation represents a degenerate conic, specifically overlapping lines forming a single line (trivial case).

Step 1: Recognize the Form

The given equation is \(4x^2 - 24x + 35 = 0\). This is a quadratic equation in one variable, which is a form of the general quadratic equation \(ax^2 + bx + c = 0\). Here, \(a = 4\), \(b = -24\), and \(c = 35\).

Step 2: Move the Constant Term

Our goal is to complete the square with the \(x\) terms. First, we move the constant term to the other side of the equation: \[ 4x^2 - 24x = -35 \]

Step 3: Factor Out the Coefficient of \(x^2\)

Factor out the 4 from the left side terms that involve \(x\):\[ 4(x^2 - 6x) = -35 \]

Step 4: Complete the Square

To complete the square, take half of the coefficient of \(x\) (which is \(-6\)), square it, and add it inside the parentheses:\[ x^2 - 6x \rightarrow (x - 3)^2 - 9 \]Thus, the equation becomes:\[ 4((x - 3)^2 - 9) = -35 \]Simplify it:\[ 4(x - 3)^2 - 36 = -35 \]

Step 5: Solve for the Standard Form

Add 36 to both sides of the equation to isolate the perfect square term:\[ 4(x - 3)^2 = 1 \]

Step 6: Divide to Simplify

Divide the entire equation by 4 to solve for \((x-3)^2\):\[ (x - 3)^2 = \frac{1}{4} \]

Step 7: Identify the Conic Section

The equation \((x - 3)^2 = \frac{1}{4}\) is in the form of \(y = k\), which is a horizontal line. In this context, since there is no \(y\) variable and the equation simplifies further just like \(x = 3\) or \(x = 3.5\), this represents a degenerate form of a pair of intersecting lines that completely overlap, simplifying to a single line at a constant x-value, hence no non-degenerate conic section.

Key Concepts

Completing the Square

Completing the square is a nifty algebraic technique used to convert a quadratic equation into a perfect square trinomial. It simplifies the equation and makes it easier to solve or analyze. To complete the square, start with a quadratic equation in the form of

• \( ax^2 + bx + c = 0 \)

First, you need to ensure the coefficient of \(x^2\) is 1. If not, factor it out from the terms involving \(x\). With our example, we started with

• \( 4(x^2 - 6x) = -35 \)

Next, focus on the expression inside the parentheses— \(x^2 - 6x\). Take half of the coefficient of \(x\) (which is

• \(-6/2 = -3\)

), square it (

• \((-3)^2 = 9\)

), and then add and subtract it within the expression to form a perfect square trinomial:

• \((x - 3)^2 - 9\)

Finally, the equation turns into a form that looks like

• \( 4((x - 3)^2 - 9) = -35 \)

This method simplifies solving the equation and leads to further identification of conic sections.

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2, and it can be expressed in the general form:

• \( ax^2 + bx + c = 0 \)

where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Quadratic equations have a specific characteristic: they form parabolas when graphically represented. However, a quadratic equation involving only one variable is considered to be a basic form. In our example,

• \( 4x^2 - 24x + 35 = 0 \)

we identified \(a = 4\), \(b = -24\), and \(c = 35\). The first step was to rearrange and prepare terms for completing the square. This is done by isolating the x-terms, transforming it into a suitable form for solving:

• \( 4x^2 - 24x = -35 \)

Understanding the basics of quadratic equations allows us to leap into solving more complex equations and recognizing various forms they can take. Subtly, this guides us into categorizing conics through their degenerate forms.

Degenerate Conic

In the world of conic sections, degenerate conics occur when a section doesn't form a typical shape like a circle, ellipse, hyperbola, or parabola. Instead, it simplifies into something else, like a single line or a point. When studying the equation:

• \( (x - 3)^2 = \frac{1}{4} \)

we discovered that it did not represent a curve. Instead, its solution simplifies to the x-value \( x = 3.5 \), which indicates a pair of lines that intersect and overlap completely to form one horizontal line. In this context, it is referred to as a degenerate conic section. The process of arriving at the form \((x - 3)^2\) shows that the equation no longer relies on another dimension (involving a y-variable, for instance), thereby reducing its representation to a simpler form. Degenerate conics are an important part of understanding the broader conic sections, highlighting what happens when conditions lead to simplifications.