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

Verified step by step guidance

1. Start with the quadratic equation given: \(x^{2} - 5x + 6 = 0\).
2. Factor the quadratic expression on the left side. Look for two numbers that multiply to \(6\) (the constant term) and add up to \(-5\) (the coefficient of \(x\)).
3. Write the factored form as \((x - a)(x - b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
4. Apply the zero-factor property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, set each factor equal to zero: \(x - a = 0\) and \(x - b = 0\).
5. Solve each equation for \(x\) to find the solutions to the quadratic equation.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.

Factoring Quadratic Expressions

Factoring 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 to apply the zero-factor property to find the roots.

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 allows solving equations like (x - 2)(x - 3) = 0 by setting each factor equal to zero.