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 \(ax^2 + bx + c = 0\), where \(a = 4\), \(b = -4\), and \(c = 1\).
2. Check if the quadratic can be factored easily. Look for two numbers that multiply to \(a \times c = 4 \times 1 = 4\) and add to \(b = -4\).
3. If factoring is possible, express the quadratic as a product of two binomials: \((mx + n)(px + q) = 0\).
4. Apply the zero-factor property, which states that if \((mx + n)(px + q) = 0\), then either \(mx + n = 0\) or \(px + q = 0\).
5. Solve each linear equation separately to find the values of \(x\).

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 make the equation true. Understanding the standard form is essential for applying solution methods.

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 to find the roots.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. This process simplifies solving the equation by enabling the use of the zero-factor property. Recognizing perfect square trinomials or using methods like grouping helps in factoring efficiently.