Find the Roots: Solving the Quadratic Equation x² - 16 = 0

To solve this problem, we'll follow these steps:

• Step 1: Recognize the structure of the equation as a difference of squares.
• Step 2: Factor the equation.
• Step 3: Use the zero-product property to find the values of $x$.

Now, let's work through each step:
Step 1: The equation given is $x^2 - 16 = 0$. This equation is a type of difference of squares, as it can be expressed in the form $a^2 - b^2 = (a - b)(a + b)$.
Step 2: Recognizing $16$ as $4^2$, we can factor the equation: $x^2 - 16 = (x - 4)(x + 4) = 0$.
Step 3: According to the zero-product property, if the product of two expressions is zero, then at least one of the expressions must be zero. Therefore, we set each factor equal to zero:
$x - 4 = 0$ or $x + 4 = 0$.
Solving these simple linear equations gives $x = 4$ and $x = -4$.

Therefore, the solutions to the equation are $x_1 = -4$ and $x_2 = 4$.