In Exercises 49–56, identify each equation without completing the square. 4x^2 - 9y^2 - 8x - 36y - 68 = 0

Verified step by step guidance

1. Identify the general form of the conic section equation: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
2. Compare the given equation $4x^2 - 9y^2 - 8x - 36y - 68 = 0$ with the general form.
3. Notice that $B = 0$, indicating there is no $xy$ term, which simplifies the identification process.
4. Observe the coefficients of $x^2$ and $y^2$: $A = 4$ and $C = -9$. Since $A$ and $C$ have opposite signs, the equation represents a hyperbola.
5. Conclude that the given equation is a hyperbola based on the signs of $A$ and $C$.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax^2 + bx + c = 0. In the context of conic sections, it can represent parabolas, ellipses, hyperbolas, or degenerate cases. Understanding the standard form and how to manipulate these equations is crucial for identifying their types.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The main types include circles, ellipses, parabolas, and hyperbolas. Each type has a specific standard form, and recognizing the coefficients and terms in a quadratic equation helps determine which conic section it represents.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, making it easier to analyze or solve. While the question specifies not to complete the square, understanding this technique is essential for identifying the conic section and rewriting the equation in standard form.