Solving for x in Terms of y: A Comprehensive Guide to Quadratic Equations

The world of algebra is rich with equations that play a crucial role in various fields including mathematics, engineering, and physics. One such equation is the quadratic equation, which can be manipulated to solve for different variables. In this guide, we will explore how to solve for x in terms of y using a given quadratic equation and employ algebraic techniques and concepts. Our focus will be on understanding the discriminant and finding valid solutions for x.

Equation Analysis

Given the equation:

3x2 - 4y2 - 8xy - 2x 1 0

We can rearrange this equation into a form that allows us to solve for y in terms of x. However, for simplicity and clarity, let's reframe the equation in terms of x, as we will be solving for x.

Step-by-Step Solution

1. Initial Equation Transformation

Let's begin with the given equation:

3x2 - 4y2 - 8xy - 2x 1 0

Rearrange the equation to group the terms involving x:

3x2 - 8xy - 2x - 4y2 1 0

2. Quadratic Formula Application

To solve for x, we will use the quadratic formula:

y (frac{8x pm sqrt{8x^2 - 4 cdot 3 cdot (3x^2 - 2x - 1)}}{2 cdot 4})

Further simplifying, we get:

y (frac{8x pm sqrt{16x^2 - 32x^2 16}}{8})

y (frac{8x pm 4sqrt{x^2 - 2x - 1}}{8})

Therefore, the solution is:

y (frac{1}{2}(2x pm sqrt{x^2 - 2x - 1}))

3. Validity of Solutions for x

For the square root to be real, the expression inside the square root must be non-negative:

x2 - 2x - 1 ≥ 0

(x - 1)2 ≥ 2

(x - 1)2 ≥ 2

x - 1 ≥ 21/2

x - 1 ≤ -21/2

Therefore, the valid solutions for x are:

x ≤ 1 - 21/2 or x ≥ 1 21/2

These solutions can be boxed for clarity:

( boxed{x le 1 - sqrt{2} text{ or } x ge 1 sqrt{2}} )

4. Determining the Value of xy

In another variant, we consider whether the given equation can be transformed to find a specific value of xy. Analyzing the given equation:

3x2 - 4y2 - 8xy - 2x 1 0

One possible transformation is: 5x2 - 4y2 - 8xy - 2x 1 0

Let's solve for this variant:

5x2 - 4y2 - 8xy - 2x 1 0

Rearranging and simplifying, we find that:

( boxed{xy 2} )

5. Solving by Quadratic Formula

Alternatively, we can solve by applying the quadratic formula directly to the original equation:

3x2 - 8xy - 2x - 4y2 1 0

Let's denote x as the unknown:

x (frac{8y pm sqrt{(8y)^2 - 4 cdot 3 cdot (4y^2 - 8xy - 1)}}{2 cdot 3})

Further simplification leads to:

x (frac{8y pm sqrt{64y^2 - 48y^2 96xy 12}}{6})

x (frac{8y pm sqrt{16y^2 96xy 12}}{6})

Therefore, the solutions are:

x (frac{8y pm 2sqrt{4y^2 24xy 3}}{6})

( boxed{x frac{4y pm sqrt{4y^2 24xy 3}}{3}} )

Conclusion

In conclusion, we have explored various methods to solve for x in terms of y using quadratic equations. The discriminant approach provides insightful solutions for valid x values. The transformation and quadratic solution approach yield a specific value of xy for a given variant. Understanding these techniques is crucial for solving complex algebraic problems.

Keywords

quadratic equations, solving for x, algebraic manipulation