Identify each equation without completing the square. 4x² - 9y² - 8x - 36y - 68 = 0

Verified step by step guidance

1. Rewrite the given equation to group the x-terms and y-terms together: \$4x^2 - 8x - 9y^2 - 36y - 68 = 0$.
2. Identify the coefficients of the squared terms: the coefficient of \(x^2\) is 4 (positive) and the coefficient of \(y^2\) is -9 (negative).
3. Since the \(x^2\) and \(y^2\) terms have opposite signs, recognize that this is the general form of a hyperbola.
4. Recall that the standard form of a hyperbola is either \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) or \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\), where the squared terms have opposite signs.
5. Therefore, without completing the square, conclude that the given equation represents a hyperbola.

Key Concepts

Identifying Conic Sections from General Quadratic Equations

Conic sections are curves obtained by intersecting a plane with a cone, represented by second-degree equations in x and y. Recognizing the type (circle, ellipse, parabola, hyperbola) involves analyzing the coefficients of x² and y², especially their signs and magnitudes, without necessarily rewriting the equation.

Discriminant of a Conic Section

The discriminant, given by B² - 4AC for the general quadratic form Ax² + Bxy + Cy² + Dx + Ey + F = 0, helps classify conics. If B=0, as in this problem, the sign of AC determines the conic: AC > 0 indicates ellipse or circle, AC = 0 parabola, and AC < 0 hyperbola.

Analyzing Coefficients Without Completing the Square

Instead of completing the square, one can identify the conic by examining the coefficients of x² and y² and their signs, along with the linear terms. This approach saves time and avoids algebraic manipulation, relying on the relationship between coefficients to classify the conic.