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

Verified step by step guidance

1. Recognize that the equation is in the form \( (x - 4)^2 = -5 \), which is suitable for applying the square root property. The square root property states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\).
2. Apply the square root property to both sides of the equation: \(x - 4 = \pm \sqrt{-5}\).
3. Since the square root of a negative number involves imaginary numbers, rewrite \(\sqrt{-5}\) as \(\sqrt{5}i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
4. Express the solutions as \(x - 4 = \pm \sqrt{5}i\).
5. Isolate \(x\) by adding 4 to both sides: \(x = 4 \pm \sqrt{5}i\).

Key Concepts

Square Root Property

The square root property states that if (x - a)^2 = b, then x - a = ±√b. This allows solving quadratic equations by isolating the squared term and taking the square root of both sides, considering both positive and negative roots.

Complex Numbers and Imaginary Unit

When the equation involves the square root of a negative number, solutions are expressed using imaginary numbers. The imaginary unit i is defined as √(-1), enabling the representation of roots of negative numbers as multiples of i.

Isolating the Variable

Before applying the square root property, the equation must be manipulated to isolate the squared expression on one side. This step ensures the equation is in the form (x - a)^2 = b, making it ready for taking square roots and solving for x.