Solving Complex Equations and Conic Sections: A Comprehensive Guide

Hello again, welcome back to our guide on solving complex equations and conic sections! In today's article, we will delve into the process of solving equations, focusing on algebraic equations and conic sections. We will start with a specific equation and then discuss the broader applications of these concepts.

How to Solve the Given Equation

Let's consider the given equation below:

The equation is:

6/(x^2 - 2) - 1/√(x^2 - 2) 2

To solve the above equation, we proceed as follows:

Simplify the equation: Let y 1/√(x^2 - 2) Substitute y into the equation: The equation becomes: 6y^2 - y - 2 0 Solve the quadratic equation: Using the quadratic formula: Discriminant D 1^2 - 4*6*(-2) 1 48 49 Solutions are: y_1 (1 7) / (2*6) 8 / 12 2/3 y_2 (1 - 7) / (2*6) -6 / 12 -1/2 Reject y_2 -1/2 as it does not fit the context. Thus, y 2/3 Substitute back y 1/√(x^2 - 2): 1/√(x^2 - 2) 2/3 Square both sides: 1/(x^2 - 2) 4/9 Multiply both sides by (x^2 - 2): x^2 - 2 9/4 Thus, x^2 9/4 2 9/4 8/4 17/4 Therefore, x ±√(17/4) ±(√17)/2

Understanding Conic Sections

Conic sections are fascinating because they represent a family of curves that can be derived by intersecting a cone with a plane. These curves include circles, ellipses, parabolas, and hyperbolas. Let's explore the concept further:

Equation of Conic Sections

Consider the general equation of conic sections:

px^2 qxy ry^2 sx ty u 0

This equation can represent a variety of conic sections depending on the coefficients:

Circle: When p q r and H 0. Ellipse: When D > 0 and Ap 2Hq Br 0. Parabola: When D 0 and Ap 2Hq Br 0. Hyperbola: When D and Ap 2Hq Br 0.

Discriminant of Conic Sections

The discriminant D plays a crucial role in determining the type of conic section:

Discriminant: D abc - 2fgh af^2 bg^2 ch^2

For conic sections:

D > 0: The conic is an ellipse (excluding the case of a circle). D 0: The conic is a parabola or a degenerate conic. D The conic is a hyperbola.

Conclusion

We have walked through the process of solving complex equations and the fundamental concepts of conic sections. Whether you're a student or a professional, understanding these concepts is crucial for tackling a wide range of mathematical challenges. If you need deeper insights, consider checking out textbooks like SK Goyal's conic sections for more detailed explanations.