(x-2y)(x+2y)=4

2 min read Jun 17, 2024

Solving the Equation (x - 2y)(x + 2y) = 4

This equation presents a simple yet interesting algebraic problem. We can approach it by understanding the pattern of the equation and using basic algebraic manipulation.

Understanding the Pattern

The left-hand side of the equation resembles the "difference of squares" pattern:

(a - b)(a + b) = a² - b²

In our case, a = x and b = 2y. So, we can rewrite the equation as:

x² - (2y)² = 4

Solving for x and y

Now, we can solve for x and y. Let's break down the steps:

Simplify: x² - 4y² = 4
Rearrange: x² = 4 + 4y²
Isolate x: x = ±√(4 + 4y²)
Simplify: x = ±2√(1 + y²)

This gives us the general solution for x in terms of y. We can substitute various values for y to find corresponding x values.

Example:

Let's say y = 1. Plugging it into our solution:

x = ±2√(1 + 1²) = ±2√2

Therefore, when y = 1, we have two possible solutions for x: x = 2√2 and x = -2√2.

Conclusion

The equation (x - 2y)(x + 2y) = 4 is solved by expressing x as x = ±2√(1 + y²). This solution highlights the importance of recognizing patterns in algebraic equations and using basic manipulation to simplify them. Furthermore, it demonstrates that there can be multiple solutions depending on the value of y.