Problem 43 - 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. $$x^{2}+4 x y+4 y^{2}-5 x-y-3=0$$

Step-by-Step Solution

Answer

1. The graph of the equation is hyperbola as the discriminant > 0. 2. The roots of the equation as obtained using the quadratic formula are \(y = \frac{1-4x \pm \sqrt{-15x^{2} + 24x - 7}}{8}\)

Step 1: Formulate the quadratic equation

The equation provided is \(x^{2}+4 x y+4 y^{2}-5 x-y-3=0\). We can rearrange the equation and put it in a form of \(a y^{2} + b y + c = 0\) where \(a, b,\) and \(c\) are functions of \(x\). Rearrange and we get \(4 y^{2} + y(4x-1) + (x^{2}-5x-3) = 0\). Here, \(a = 4, b = 4x-1,\) and \(c = x^{2}-5x-3\).

Step 2: Find the discriminant

The discriminant, \(D\) of a quadratic equation \(a y^{2} + b y + c = 0\) is given by \(D = b^{2} - 4*a*c \). Substituting \( a = 4, b = 4x-1,\) and \(c = x^{2}-5x-3\) we get \(D = (4x-1)^{2} - 4*4*(x^{2}-5x-3)\). We simplify the equation to get \(D = -15x^{2} + 24x - 7\). The discriminant > 0 that means the graph of the equation is a hyperbola.

Step 3: Solving for y using quadratic formula

The quadratic formula is given by \(y = \frac{-b \pm \sqrt{D}}{2a} \). Substituting \(a=4\), \(b=4x-1\) and \(D = -15x^{2} + 24x - 7\) into the equation, we get \(y = \frac{1-4x \pm \sqrt{-15x^{2} + 24x - 7}}{8}\).

Step 4: Graphing the equation

This step requires us to graph the function using a graphing utility. This would visually represent the solution we obtained.

Key Concepts

Discriminant

The discriminant is an integral part of the quadratic equation, helping to determine the nature of the roots without solving the equation entirely. To find the discriminant, use the formula \(D = b^2 - 4ac\). In the given exercise, the quadratic equation is expressed as \(4y^2 + y(4x - 1) + (x^2 - 5x - 3) = 0\), where \(a = 4\), \(b = 4x - 1\), and \(c = x^2 - 5x - 3\).

By substituting these values into the discriminant formula, we find \(D = (4x - 1)^2 - 4 \times 4 \times (x^2 - 5x - 3)\). Upon simplifying, \(D\) becomes \(-15x^2 + 24x - 7\). A positive discriminant indicates that the quadratic equation has two distinct real roots, signifying that our original equation graph is a hyperbola. Knowing how to calculate and interpret the discriminant is crucial as it provides insight into the type of conic section represented by the equation.

Quadratic Formula

The quadratic formula is a straightforward method to find the solutions or roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is written as \(y = \frac{-b \pm \sqrt{D}}{2a}\), where \(D\) represents the discriminant, \(b^2 - 4ac\).

With the problem given, where \(a = 4\), \(b = 4x - 1\), and the discriminant we found is \(D = -15x^2 + 24x - 7\), inserting these values into the quadratic formula yields the solutions for \(y\):

• \(y = \frac{1 - 4x + \sqrt{-15x^2 + 24x - 7}}{8}\)
• \(y = \frac{1 - 4x - \sqrt{-15x^2 + 24x - 7}}{8}\)

These solutions give the \(y\)-coordinates for given \(x\)-values, necessary for sketching the graph of the equation. The quadratic formula is an invaluable tool as it directly ties into finding where our equation crosses the \(y\)-axis, which is critical for graph interpretation.

Graphing Utility

Graphing utilities are digital tools used for sketching the graphs of equations. They greatly simplify the process by allowing us to visualize complex equations quickly and accurately. For the given equation \(x^{2}+4xy+4y^{2}-5x-y-3=0\), a graphing utility provides a clear illustration of the conic section type — in this case, a hyperbola due to the positive discriminant.

When using a graphing utility:

• Input the equation after rearranging it, if necessary, to suit the tool’s requirements.
• Check the graph for intercepts, especially for accuracy and correctness of the solutions previously found using algebraic methods like the quadratic formula.
• Observe the general shape, direction, and orientation of the graph to gain insights into real-world phenomena the equation may model.

Graphing utilities transform abstract mathematical concepts into visible curves, circles, and lines, making the analysis of relationships and behaviors much more intuitive.