Solve each quadratic equation using the zero-factor property. See Example 5. 4x² - 4x + 1 = 0

Verified step by step guidance

1. Recognize that the given equation is a quadratic equation in the form \$4x^{2} - 4x + 1 = 0$.
2. Try to factor the quadratic expression on the left side. Look for two binomials \((ax + b)(cx + d)\) such that when multiplied, they give \$4x^{2} - 4x + 1$.
3. Once factored, the equation will look like \((2x - 1)(2x - 1) = 0\) or \((2x - 1)^{2} = 0\).
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. So, set each factor equal to zero: \$2x - 1 = 0$.
5. Solve the resulting linear equation for \(x\) by isolating \(x\): add 1 to both sides and then divide by 2, giving \(x = \frac{1}{2}\).

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed, and solving it means finding 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 and setting each factor equal to zero.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. For example, 4x² - 4x + 1 can be factored into (2x - 1)(2x - 1). This step is essential before applying the zero-factor property to find the roots.