Problem 43 - Precalculus with Limits

Question

In Exercises 43-50, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for \(y\), and (c) use a graphing utility to graph the equation. \(16x^2-8xy+y^2-10x+5y=0\)

Step-by-Step Solution

Answer

The graph of the equation will be an ellipse as indicated by a negative discriminant. There are two possible solutions for \(y\) after applying the quadratic formula. The graph generated by the equation will confirm this

Step 1: Classification of the graph using the Discriminant

Rewrite the equation in a 'general form' of a quadratic as follows: \((8x - y)^2 = 10x - 5y\). The equation takes a form of \(Ax^2+Bx+C = 0\), \(A = 16, B = -5\) and \(C = 10\). The discriminant of a quadratic equation \(Ax^2+Bx+C = 0\) is given by \(B^2-4AC\). Substituting the values of \(A, B, C\), into the formula gives \((-5)^2 - 4(16)(10) = - 630\). The discriminant is less than zero (\(< 0\)) indicating that the graph will be an ellipse.

Step 2: Solve for \(y\) using the quadratic formula

The quadratic formula for an equation \(Ax^2+Bx+C = 0\) is \(x = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\). To find the values of \(y\), let's take our rewritten equation \((8x - y)^2 = 10x - 5y\), rearranged to \(8x^2 - 16xy + y^2 - 10x + 5y = 0\). The coefficients of \(y\) are \(A = 1, B = 16x - 5, C = 8x^2 - 10x\). \(y = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\), thus substituting the values of \(A, B, C\) into the formula gives \(y = \frac{(5 - 16x) \pm \sqrt{(16x - 5)^2 - 4(8x^2-10x)}}{2}\), simplifying this gives two solutions for \(y\).

Step 3: Graph the equation

Using a graphing utility, the equation \((8x - y)^2 = 10x - 5y\) would be plotted. As the discriminant indicated, the graph will be an ellipse.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations of the form \(Ax^2 + Bx + C = 0\). It is given by the formula:

• \(x = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\)

This formula provides solutions for \(x\) by using the coefficients \(A\), \(B\), and \(C\) from the equation.

In problems like the one given, we may need to reformulate the equation so that we can apply the quadratic formula to solve for \(y\). This involves carefully selecting and rearranging terms to fit the standard quadratic form.
The discriminant, \(B^2 - 4AC\), found under the square root, determines the nature of the roots:

• If it is positive, there are two distinct real solutions.
• If it is zero, there is one real solution, indicating that the solutions are equal.
• If it is negative, the solutions are complex or imaginary.

In our exercise, the discriminant helps us validate our expectation of the graph being an ellipse, as a negative discriminant suggested.

Graphing Utility

A graphing utility is a tool, either a software application or a graphing calculator, that helps plot equations visually. It's especially handy with complex equations, such as the one in this exercise, where manually plotting might be cumbersome.

• Graphing utilities allow us to analyze the shape, intercepts, and behavior of an equation.
• They offer options to zoom in, zoom out, and adjust viewing windows for better visualization of curves or intersections.
• We can also use them to check the solutions we calculated using the quadratic formula or other algebraic methods.

When applied to our exercise, a graphing utility confirms that the equation \((8x - y)^2 = 10x - 5y\) does form an ellipse, as predicted by examining the discriminant.

Ellipse

An ellipse is a type of conic section formed by the intersection of a plane with a cone. Unlike a circle, an ellipse has two focal points and is elongated along one axis.

In mathematical terms, an ellipse often takes the form:

• \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)

where \((h, k)\) is the center, and \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.

In the context of the given equation and its discriminant yielding \(-630\), this confirmed the graph is an ellipse since the discriminant was less than zero.
Properties of ellipses include:

• Symmetrical about both the horizontal and vertical axes through the center.
• Each point on an ellipse is such that the sum of the distances from the two focal points is constant.

Understanding these properties helps in visualizing how the calculated solutions relate to the actual graph.