Solve each equation using the zero-factor property. See Example 1. x^2 - 64 = 0

Verified step by step guidance

1. Recognize that the equation $x^2-64=0$ is a difference of squares.
2. Recall the difference of squares formula: $a^2-b^2=(a-b)(a+b)$.
3. Identify $a^2=x^2$ and $b^2=64$, so $a=x$ and $b=8$.
4. Rewrite the equation as $(x-8)(x+8)=0$.
5. Apply the zero-factor property: set each factor equal to zero, $x-8=0$ and $x+8=0$, and solve for $x$.

Key Concepts

Zero-Factor Property

The Zero-Factor Property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is essential for solving polynomial equations, as it allows us to set each factor equal to zero to find the solutions. For example, if we have an equation like (x - 4)(x + 4) = 0, we can conclude that x - 4 = 0 or x + 4 = 0.

Factoring Quadratic Equations

Factoring quadratic equations involves rewriting the equation in a product form, typically as (x - p)(x + q) = 0, where p and q are the roots of the equation. In the case of x^2 - 64 = 0, it can be factored as (x - 8)(x + 8) = 0. This step is crucial for applying the Zero-Factor Property effectively.

Solving for x

Once the equation is factored, solving for x involves setting each factor equal to zero and solving for the variable. For the equation (x - 8)(x + 8) = 0, we set x - 8 = 0 and x + 8 = 0, leading to the solutions x = 8 and x = -8. This process is fundamental in finding the roots of the original equation.