Solve each equation using the zero-factor property. See Example 1. x^2 - 5x + 6 = 0

Verified step by step guidance

1. Identify the quadratic equation: \(x^2 - 5x + 6 = 0\).
2. Factor the quadratic expression on the left-hand side. Look for two numbers that multiply to 6 (the constant term) and add to -5 (the coefficient of the linear term).
3. The numbers that satisfy these conditions are -2 and -3. Therefore, factor the quadratic as \((x - 2)(x - 3) = 0\).
4. Apply the zero-factor property, which states that if a product of factors equals zero, then at least one of the factors must be zero.
5. Set each factor equal to zero: \(x - 2 = 0\) and \(x - 3 = 0\), and solve each equation 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 quadratic equations, as it allows us to set each factor equal to zero to find the solutions of the equation.

Factoring Quadratic Equations

Factoring quadratic equations involves rewriting the equation in the form of a product of two binomials. For example, the equation x^2 - 5x + 6 can be factored into (x - 2)(x - 3) = 0. This step is crucial for applying the Zero-Factor Property effectively.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Understanding the standard form of quadratic equations is vital for recognizing how to apply factoring and the Zero-Factor Property to find the roots of the equation.