Solve each quadratic equation using the zero-factor property. See Example 5. x² - 100 = 0

Verified step by step guidance

1. Rewrite the equation in the form \(x^2 - 100 = 0\) to identify it as a difference of squares.
2. Recognize that \(x^2 - 100\) can be factored using the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), where \(a = x\) and \(b = 10\).
3. Factor the equation as \((x - 10)(x + 10) = 0\).
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: \(x - 10 = 0\) and \(x + 10 = 0\).
5. Solve each equation separately to find the solutions for \(x\): \(x = 10\) and \(x = -10\).

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 quadratic equations factored into binomials, allowing us to set each factor equal to zero and solve for the variable.

Factoring Quadratic Equations

Factoring involves rewriting a quadratic equation as a product of two binomials or simpler expressions. For example, x² - 100 can be factored using the difference of squares into (x - 10)(x + 10). Factoring simplifies the equation and prepares it for applying the zero-factor property.

Difference of Squares

The difference of squares is a special factoring formula: a² - b² = (a - b)(a + b). Recognizing this pattern helps factor expressions like x² - 100, where 100 is 10². This technique is crucial for breaking down certain quadratic equations quickly and accurately.