Solve each equation using the zero-factor property. See Example 1. 9x^2 - 12x + 4 = 0

Verified step by step guidance

1. First, recognize that the equation is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, \( a = 9 \), \( b = -12 \), and \( c = 4 \).
2. Next, try to factor the quadratic expression \( 9x^2 - 12x + 4 \). Look for two numbers that multiply to \( a \times c = 9 \times 4 = 36 \) and add to \( b = -12 \).
3. The numbers that satisfy these conditions are \(-6\) and \(-6\). Rewrite the middle term \(-12x\) using these numbers: \( 9x^2 - 6x - 6x + 4 \).
4. Group the terms to factor by grouping: \((9x^2 - 6x) + (-6x + 4)\).
5. Factor out the greatest common factor from each group: \(3x(3x - 2) - 2(3x - 2)\). Notice that \((3x - 2)\) is a common factor, so factor it out: \((3x - 2)(3x - 2) = 0\).

Key Concepts

Zero-Factor Property

The Zero-Factor Property states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This principle is essential for solving polynomial equations, as it allows us to set each factor equal to zero to find the solutions of the equation.

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 structure of quadratic equations is crucial for applying the zero-factor property, as it often involves factoring the quadratic into two binomials.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For quadratic equations, this often means expressing the equation in a form like (px + q)(rx + s) = 0. Mastery of factoring techniques is vital for effectively using the zero-factor property to solve equations.