Problem 39 - 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. $$12 x^{2}-6 x y+7 y^{2}-45=0$$

Step-by-Step Solution

Answer

The given equation represents an Ellipse. The exact value of \(y\) depends on the value of \(x\). Using a graphing utility, we can visually see the ellipse.

Step 1: Compute the Discriminant

Rewrite the equation in general form \( A x^{2}+ B x y + C y^{2} + D = 0 \). From this, we will calculate the discriminant using the formula \(B^2 - 4AC\) which will help in classifying the graph. So, we identify \(A=12\), \(B=-6\), \(C=7\). Thus, the Discriminant, \(D=B^{2}-4 A C=(-6)^{2}-4(12)(7) = -192.\)

Step 2: Classify the Graph

Since the Discriminant is less than zero (D<0), the graph of the given equation is an Ellipse.

Step 3: Solve for \(y\) using Quadratic Formula

The Quadratic formula is used to find the roots of a quadratic equation. It is\( y = \frac{- B \pm \sqrt{B^{2}-4 A(C - x^{2})}}{2C} \). Given the equation is \(12 x^{2}-6xy+7 y^{2} - 45 = 0\), solving for \(y\), we get \(y = \frac{6x \pm \sqrt{(-6x)^{2}-4.7.(12x^{2}- 45)}}{14}\). You can simplify this expression further. However, the exact value of \(y\) depends on the value of \(x\).

Step 4. Graph the Equation Using a Graphing Utility

For this step a graphing utility software is required. Plug the equation \(12 x^{2}-6xz+7 z^{2}-45=0\) into the graphing utility to visualize the graph. Since the Discriminant was less than zero, we expect to see an ellipse. The graphing utilities can give a visual confirmation of this.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x with a nonzero coefficient for the x² term. It has the standard form of ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Quadratic equations are fundamental to algebra and are characterized by the highest power of the unknown being two. Solving such equations often involves finding the values of x that make the equation true, known as the 'roots' or 'solutions' of the equation.

When graphing a quadratic equation, you typically get a parabola. However, when the equation involves both x and y terms, as in the exercise 12x² - 6xy + 7y² - 45 = 0, we are describing a conic section rather than a traditional single-variable parabola, which can lead to different types of curves such as ellipses, hyperbolas, or circles depending on the coefficients and their interrelation.

Graphing Quadratic Equations

Graphing quadratic equations helps to visualize the set of all solutions (x,y) that satisfy the equation. When an equation is in two variables, like the given exercise, it represents a conic section rather than a simple parabola. The discriminant in these cases tells us about the nature of the graph without actually plotting it: a positive discriminant corresponds to a hyperbola, zero discriminant to parabolas, and a negative discriminant indicates an ellipse or a circle.

Using graphing software or a graphing calculator, you input the quadratic equation to obtain its graph. This practice can enhance understanding by illustrating properties like symmetry, intercepts, and the orientation of the graph, which are not immediately apparent from the equation itself.

Quadratic Formula

Solving for y in a quadratic equation involving both x and y requires an adaptation of the quadratic formula. The traditional formula, y = (-b ± √(b²-4ac)) / (2a), finds the roots of a quadratic equation ax² + bx + c = 0. However, when applying this to equations with a mixed term xy, the formula must be accommodated for the presence of x values.

In our exercise, we adjust it to y = (6x ± √((-6x)²-4·7·(12x²-45))) / 14, which incorporates the coefficients of the terms involving y and the changes in the constant term based on the value of x. It's essential to consider x as a constant when solving for y and vice versa.

Ellipse Classification in Quadratic Equations

In the exercise, we're faced with a quadratic equation that includes both x and y terms. The classification of the resulting graph as an ellipse comes from the discriminant, which is the part of the equation under the square root in the quadratic formula. In our case, the calculated discriminant is negative (-192), a clear indicator that the conic section represented by the equation is an ellipse.

It's this discriminant value that distinguishes an ellipse from other conic sections. For an ellipse in standard position, the quadratic equation will not have an xy term, and the coefficients of x² and y² will be positive but not necessarily equal (if they were equal, we would have a circle). The general equation can be transformed to make the classification more apparent, but the discriminant gives us a quick tool for classification without the need for transforming or completing the square.