Problem 42 - Precalculus

Question

(a) use the discriminant to classify the graph of the equation, (b) use the Quadratic Formula to solve for \(y\) and (c) use a graphing utility to graph the equation. $$36 x^{2}-60 x y+25 y^{2}+9 y=0$$

Step-by-Step Solution

Answer

The graph of the equation is a parabola. The equation solved for \(y\) in terms of \(x\) is \(y = \frac{-60 \pm \sqrt{3600 + 3600x^{2}}}{50}\). Consider using a graphing utility to visually verify these results.

Step 1: Identify discriminant

The equation in the exercise can be written in the general form \(Ax^{2} + Bxy + Cy^{2} + Dy = 0\). We have \(A = 36\), \(B = -60\), \(C = 25\) and \(D = 9\). The discriminant is given by \(B^2 - 4AC = (-60)^2 - 4*36*25 = 3600 - 3600 = 0\).

Step 2: Classify graph using discriminant

Since the discriminant is equal to zero, this means that the equation represents a parabola.

Step 3: Solve for \(y\) using the Quadratic formula

In order to solve for \(y\), first re-arrange the equation into the form \(ay^{2} + by + c = 0\), which yields \(25y^{2} + 60y - 36x^{2} = 0\). Apply the Quadratic formula to solve for \(y\), which gives:\[y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} = \frac{-60 \pm \sqrt{(60)^{2} - 4(25)(-36x^{2})}}{2*25}\]

Step 4: Simplify the solution

Simplifying the above equation provides the solution for \(y\) in terms of \(x\), \(y = \frac{-60 \pm \sqrt{3600 + 3600x^{2}}}{50}\)

Step 5: Graph Equation Using a Graphing Utility

Lastly, to graph the equation using a graphing utility, simply input the equation \(y = \frac{-60 \pm \sqrt{3600 + 3600x^{2}}}{50}\), and the program will generate a graph for the parabola.

Key Concepts

Discriminant

The concept of the discriminant provides valuable insight into the nature of conic sections equations. In the context of the given exercise, the discriminant is associated with the general conic equation format, which includes the terms of the form \(Ax^{2} + Bxy + Cy^{2} + Dy = 0\). Here, the discriminant formula \(B^2 - 4AC\) helps us determine the type of conic section the graph of the equation represents.
To find the discriminant:

• Substitute the respective values of \(A\), \(B\), and \(C\) from the equation into the formula.
• Perform the calculations: \((-60)^2 - 4 \times 36 \times 25 = 3600 - 3600 = 0\).
• The result, \(0\), indicates that the graph is a parabola.

The zero result tells us that the conic section is degenerate, forming a special type of parabola rather than the usual oval shapes like ellipses or hyperbolas. Understanding the discriminant can give you a quick glance at the shape of the graph without further calculations.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations of the form \(ay^{2} + by + c = 0\). In this exercise, the goal is to solve for \(y\), treating all other terms as constants or known values.
Here's how it's applied:

• The original equation is rearranged into the quadratic format related to \(y\), yielding \(25y^{2} + 60y - 36x^{2} = 0\).
• Applying the quadratic formula \(y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) allows us to substitute \(a = 25\), \(b = 60\), and \(c = -36x^2\).
• This gives \(y = \frac{-60 \pm \sqrt{3600 + 3600x^{2}}}{50}\), showing \(y\) expressed in terms of \(x\).

The formula's "plus-minus" component \(\pm\) indicates two possible values for \(y\), which is typical for parabolas. The outcomes demonstrate how algebraic methods can be used to express one variable in relation to another.

Graphing Utility

A graphing utility is an essential tool for visualizing mathematical equations. It allows you to easily see the shape and intersection of graphs through digital plotting. For the given exercise, a graphing utility like a calculator or computer software can be used to graph the equation.
To graph using a utility:

• Enter the derived equation for \(y\), which is \(y = \frac{-60 \pm \sqrt{3600 + 3600x^{2}}}{50}\).
• Choose "Plot" or "Graph" from the options provided in the software.
• The utility will generate a visual representation of the graph.

This graph helps in understanding the geometric nature of the problem, offering a visual confirmation of the algebraic findings like it being a parabola. It also supports further exploration by allowing the user to adjust parameters or constraints and observe how the graph changes.