Solve each equation using the zero-factor property. 9x² - 12x + 4 = 0

Verified step by step guidance

1. Recognize that the equation \$9x^2 - 12x + 4 = 0\( is a quadratic equation and try to factor it into the form \)(ax + b)(cx + d) = 0$.
2. Look for two binomials whose product gives the quadratic: find numbers that multiply to \$9 \times 4 = 36\( and add to \)-12$ to help with factoring.
3. Rewrite the middle term \(-12x\) using the two numbers found, then group terms to factor by grouping.
4. After factoring by grouping, express the quadratic as a product of two binomials equal to zero, i.e., \((3x - 2)(3x - 2) = 0\) or \((3x - 2)^2 = 0\).
5. Apply the zero-factor property: set each factor equal to zero and solve for \(x\), so solve \$3x - 2 = 0$ to find the solution(s).

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 property is essential for solving polynomial equations by factoring, as it allows us to set each factor equal to zero and solve for the variable.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials or other factors. For the equation 9x² - 12x + 4 = 0, factoring helps break down the quadratic into simpler expressions that can be set to zero using the zero-factor property.

Solving Quadratic Equations

Solving quadratic equations means finding the values of the variable that satisfy the equation. After factoring, each factor is set equal to zero, and solving these linear equations yields the roots or solutions of the original quadratic.