Solve each quadratic equation using the zero-factor property. See Example 5.


9x² - 12x + 4 = 0

Verified step by step guidance

1. Recognize that the given equation is a quadratic equation in the form \$9x^{2} - 12x + 4 = 0$.
2. Try to factor the quadratic expression on the left side. Look for two binomials of the form \((ax + b)(cx + d)\) whose product equals \$9x^{2} - 12x + 4$.
3. Use the zero-factor property, which states that if \((ax + b)(cx + d) = 0\), then either \(ax + b = 0\) or \(cx + d = 0\).
4. Set each factor equal to zero and solve for \(x\): solve \(ax + b = 0\) and \(cx + d = 0\) separately.
5. Write the solutions for \(x\) obtained from each equation as the roots of 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 its solutions are the values of x that satisfy the equation.

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 property is used to solve quadratic equations by factoring them into binomials set equal to zero.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. For example, 9x² - 12x + 4 can be factored into (3x - 2)(3x - 2). This step is essential before applying the zero-factor property to find the roots.