Solve each equation using the square root property. See Example 2. (x - 4)^2 = -5

Verified step by step guidance

1. Recognize that the equation \((x - 4)^2 = -5\) is in the form \((a)^2 = b\), where \(a = x - 4\) and \(b = -5\).
2. Apply the square root property, which states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\).
3. Substitute \(a = x - 4\) and \(b = -5\) into the square root property to get \(x - 4 = \pm \sqrt{-5}\).
4. Recognize that \(\sqrt{-5}\) involves the square root of a negative number, which introduces the imaginary unit \(i\), where \(i = \sqrt{-1}\).
5. Express \(\sqrt{-5}\) as \(\sqrt{5}i\), leading to the solutions \(x - 4 = \pm \sqrt{5}i\).

Key Concepts

Square Root Property

The square root property states that if x^2 = k, then x = ±√k. This property is essential for solving quadratic equations, particularly when the equation is in the form of a perfect square. It allows us to isolate the variable by taking the square root of both sides, leading to two possible solutions.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where i is the imaginary unit defined as √(-1). When solving equations that yield negative values under the square root, such as in this case, complex numbers become necessary to express the solutions.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. Understanding the structure of quadratic equations is crucial for applying the square root property effectively, as it helps identify when the equation can be simplified to a form suitable for solving.