Solving Equations Transformable to Quadratic Form: A Detailed Guide

Have you ever encountered an equation that seems more complicated than a straightforward linear one? In this article, we will guide you through the process of transforming and solving equations that can be brought to the form of a quadratic equation. These types of equations are fundamental in algebra and are applicable in various real-world scenarios, including physics, engineering, and economics.

Introduction to Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax^2 bx c 0. These equations play a crucial role in various mathematical and practical applications. By understanding how to solve them, you can tackle a wide range of problems, from calculating areas and volumes to analyzing motion and optimization problems.

Transformation to Quadratic Form

In the given problem, we need to solve the equation 2/x - 3 x/2 -1/21. Let's break down the steps to transform this equation into a quadratic form and then solve it.

Step 1: Simplify the Equation

The given equation is:

2/x - 3 x/2 -1/21

To simplify, we find a common denominator for the terms on the left side:

4 - 6x x^2 -2x

Step 2: Bring All Terms to One Side

The next step is to bring all terms to one side, making the right side equal to zero:

x^2 - 4x 4 0

Step 3: Solve the Quadratic Equation

This quadratic equation can be factored as:

(x - 2)^2 0

From this, we find that:

x 2

Therefore, the solution to the equation is:

x 2

Another Example: 2/(x - 3) x/2 -1/2

Let's consider another example to further illustrate the process. The equation 2/(x - 3) x/2 -1/2 can be simplified as follows:

Step 1: Find a Common Denominator

Multiply both sides by the common denominator, which is:

2(x - 3)

This results in:

4 x(x - 3) -1(x - 3)

Expanding and simplifying:

x^2 - 3x 4 -x 3

Moving all terms to one side:

x^2 - 2x 1 0

Step 2: Solve the Quadratic Equation

This quadratic equation can be solved by factoring:

(x - 1)^2 0

Thus, the solution is:

x 1

Another Example: 2/x - 3 x/2 -1/2

For yet another example, let's consider the equation 2/x - 3 x/2 -1/2. In this case, we assume that the term should be 2/x - 3 x/2 -1/2.

Step 1: Find a Common Denominator

Multiplying both sides by 2x (the common denominator) gives:

4 - 6x x^2 -2x

Rearranging the terms:

x^2 - 4x 4 0

Factoring the quadratic equation:

(x - 2)^2 0

The solution is:

x 2

Additionally, another method using the quadratic formula:

x (5 ± √(25 - 16))/2

This simplifies to:

x 4 or x 1

Conclusion

In conclusion, solving equations that can be transformed to a quadratic form is a crucial skill in algebra. By understanding the steps involved, you can simplify and solve complex equations. The examples provided demonstrate the process step-by-step, making it easier to tackle similar problems in various fields. Whether it's in academic settings or practical applications, mastering quadratic equations can significantly enhance problem-solving skills.

Keywords: quadratic equation, solving equations, algebraic manipulation, transformations