Solve each equation using the zero-factor property. x² - 64 = 0

Verified step by step guidance

1. Recognize that the equation \(x^2 - 64 = 0\) is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).
2. Rewrite the equation as \((x)^2 - (8)^2 = 0\) to identify \(a = x\) and \(b = 8\).
3. Factor the left side using the difference of squares formula: \((x - 8)(x + 8) = 0\).
4. Apply the zero-factor property, which states that if a product of two factors equals zero, then at least one of the factors must be zero. Set each factor equal to zero: \(x - 8 = 0\) and \(x + 8 = 0\).
5. Solve each equation separately to find the solutions for \(x\): \(x = 8\) and \(x = -8\).

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 property is essential for solving polynomial equations by factoring, as it allows us to set each factor equal to zero and solve for the variable.

Factoring Difference of Squares

The difference of squares is a special factoring pattern where an expression of the form a^2 - b^2 can be factored into (a - b)(a + b). Recognizing this pattern helps simplify equations like x^2 - 64 = 0 into (x - 8)(x + 8) = 0, making it easier to apply the zero-factor property.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of the variable that satisfy the equation. After factoring, each factor is set to zero, and solving these linear equations yields the roots or solutions of the quadratic equation.