(x-a)(x+a)=8

2 min read Jun 17, 2024

Solving the Equation: (x - a)(x + a) = 8

This equation presents a unique challenge because it involves two variables, x and a. To solve it effectively, we need to understand the concepts of factoring and quadratic equations.

Recognizing the Pattern

The left-hand side of the equation is a special product: the difference of squares. We can see this by recognizing the pattern:

(x - a)(x + a) = x² - a²

Therefore, our original equation can be rewritten as:

x² - a² = 8

Solving for x

To solve for x, we need to isolate it. Here's how we can proceed:

Transpose the constant term: x² = a² + 8
Take the square root of both sides: x = ±√(a² + 8)

Understanding the Solution

This solution tells us that x can have two possible values, depending on the value of a:

Positive value: x = √(a² + 8)
Negative value: x = -√(a² + 8)

Example:

Let's assume a = 3.

Substituting this value into the equation for x, we get:

x = √(3² + 8) = √17
x = -√(3² + 8) = -√17

Therefore, when a = 3, the equation has two solutions: x = √17 and x = -√17.

Conclusion

Solving the equation (x - a)(x + a) = 8 requires understanding the difference of squares pattern and isolating x using algebraic manipulations. The solution for x will depend on the value of a, and the equation will generally have two solutions.