(x-a)(x+a)=12

3 min read Jun 17, 2024

Solving the Equation (x - a)(x + a) = 12

This equation is a quadratic equation in disguise, and it can be solved using a few different methods. Let's explore them:

Expanding and Simplifying

Expand the product:
(x - a)(x + a) = x² - a²
Set up the equation: x² - a² = 12
Rearrange into standard quadratic form: x² - a² - 12 = 0

Now, we have a quadratic equation in the form of ax² + bx + c = 0, where a = 1, b = 0, and c = -a² - 12.

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values:

x = [0 ± √(0² - 4 * 1 * (-a² - 12))] / 2 * 1

x = ± √(4a² + 48) / 2

x = ± √(a² + 12)

Therefore, the solutions to the equation are:

x = √(a² + 12) and x = -√(a² + 12)

Understanding the Solutions

The solutions depend on the value of 'a'. If 'a' is a real number, the solutions will also be real numbers.
The equation represents a hyperbola. The graph of (x - a)(x + a) = 12 is a hyperbola centered at (a, 0) with a vertical axis of symmetry. The solutions represent the x-intercepts of the hyperbola.

Example

Let's say a = 3. Then the solutions would be:

x = √(3² + 12) = √21 x = -√(3² + 12) = -√21

This means that when a = 3, the hyperbola intersects the x-axis at x = √21 and x = -√21.

Conclusion

The equation (x - a)(x + a) = 12 is a quadratic equation that can be solved using the quadratic formula. The solutions depend on the value of 'a' and represent the x-intercepts of the hyperbola defined by the equation.