Solving Complex Quadratic Equations: Techniques and Applications

Quadratic equations are an essential part of algebra and mathematics in general. They are often encountered in various fields such as physics, engineering, and economics. This article delves into solving complex quadratic equations, providing techniques and applications.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, typically represented as ax^2 bx c 0 where a, b, and c are constants, and ane;0. Solving these equations is fundamental in understanding their behavior and applications.

Problem Analysis and Setup

Let's consider the given equation:

x^2 - 1x - 4x^2 - 4x 4 7^2

After simplifying, we get:

(x^2 - 5x - 4) (x^2 4x - 1) 7^2

Simplifying the Equation

To solve this, we need to simplify it step by step:

1. x^2 - 5x - 4 7^2 2. x^2 4x - 1 7^2

Substitution and Factoring

Let's use substitution to simplify the problem:

Let y x^2 - 4. This gives us:

y - 5y - 2^2 7^2

Rewriting, we get:

y^2 - 5y - 9^2 0

Further simplification using polynomial factoring:

y^2 - 9y - 1y - 9^2 0

This can be factored as:

(y - 9)(y 1) 0

Solving for y

From the factored equation:

y 9, y -1

Solving for x

Now, substituting back:

1. For y 9 and 9 x^2 - 4, we get:

2. For y -1 and -1 x^2 - 4, this does not give real solutions:

Solving these, we get:

1. x^2 13 gives x ±sqrt{13} 2. x^2 3 gives x ±sqrt{3}

General Solution

The general solutions to the equation are:

x 5 ± sqrt{21} ± frac{9 ± sqrt{65}}{2}

This gives us four roots:

x 5 - sqrt{21} - frac{9 - sqrt{65}}{2}

x 5 sqrt{21} - frac{9 sqrt{65}}{2}

Conclusion

Solving complex equations requires a systematic approach. By substituting and factoring, we can simplify the equation and find its roots. This method can be applied to similar equations in various fields.