Solve each equation in Exercises 15–34 by the square root property.(x + 3)^2 = - 16

Verified step by step guidance

1. Recognize that the equation \((x + 3)^2 = -16\) involves a perfect square on the left side.
2. Recall the square root property, which states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\).
3. Apply the square root property to the equation: \(x + 3 = \pm \sqrt{-16}\).
4. Recognize that \(\sqrt{-16}\) involves the square root of a negative number, which introduces the imaginary unit \(i\), where \(i = \sqrt{-1}\).
5. Rewrite \(\sqrt{-16}\) as \(4i\), so the equation becomes \(x + 3 = \pm 4i\).

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, as it allows us to isolate the variable by taking the square root of both sides. It is particularly useful when the equation is in the form of a perfect square, enabling us to find both positive and negative solutions.

Complex Numbers

Complex numbers extend the real number system to include solutions to equations that do not have real solutions, such as the square root of negative numbers. A complex number is expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as √(-1). Understanding complex numbers is crucial when dealing with equations that yield negative results under the square root.

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve equations. It involves rearranging the equation to get the variable on one side and all other terms on the opposite side. In the context of the square root property, this often means first simplifying the equation to a standard form before applying the property to find the solutions.