Solve each equation using the zero-factor property. x² - 5x + 6 = 0

Verified step by step guidance

1. Start with the given quadratic equation: \(x^2 - 5x + 6 = 0\).
2. Factor the quadratic expression on the left side. Look for two numbers that multiply to \(6\) and add up to \(-5\).
3. Write the factored form as a product of two binomials: \((x - a)(x - b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
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: \(x - a = 0\) and \(x - b = 0\).
5. Solve each simple equation for \(x\) to find the solutions to the original quadratic equation.

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 used to solve equations by factoring expressions and setting each factor equal to zero to find the solutions.

Factoring Quadratic Equations

Factoring quadratic equations involves rewriting a quadratic expression as a product of two binomials. For example, x² - 5x + 6 factors into (x - 2)(x - 3). This step is essential before applying the zero-factor property to solve the equation.

Solving Quadratic Equations

Solving quadratic equations means finding the values of the variable that satisfy the equation. After factoring, setting each factor equal to zero and solving for the variable gives the roots or solutions of the quadratic equation.