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 \(ax^{2} + bx + c = 0\), where \(a = 9\), \(b = -12\), and \(c = 4\).
2. Check if the quadratic can be factored easily. Notice that \$9x^{2} - 12x + 4\( is a perfect square trinomial because \)9x^{2} = (3x)^{2}\( and \)4 = 2^{2}\(, and the middle term \)-12x$ equals \(2 \times 3x \times (-2)\).
3. Rewrite the quadratic as a square of a binomial: \((3x - 2)^{2} = 0\).
4. Apply the zero-factor property, which states that if a product of factors equals zero, then at least one of the factors must be zero. Here, set the factor equal to zero: \$3x - 2 = 0$.
5. Solve the linear equation \$3x - 2 = 0\( by isolating \)x\(: add 2 to both sides and then divide both sides by 3 to find the solution for \)x$.

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. This step is essential before applying the zero-factor property, as it breaks down the quadratic into simpler expressions that can be individually set to zero.