Solve each equation in Exercises 15–34 by the square root property. (x - 3)^2 = - 5

Verified step by step guidance

1. Recognize that the equation \((x - 3)^2 = -5\) involves a squared term set equal to a negative number.
2. Recall that the square root property states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\).
3. Notice that \(-5\) is negative, and the square root of a negative number involves imaginary numbers.
4. Rewrite the equation as \(x - 3 = \pm \sqrt{-5}\), which implies \(x - 3 = \pm i\sqrt{5}\) since \(i\) is the imaginary unit.
5. Solve for \(x\) by adding 3 to both sides, resulting in \(x = 3 \pm i\sqrt{5}\).

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 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). In the context of the given equation, the presence of a negative value under the square root indicates that the solutions will involve complex numbers, as the square root of a negative number is not defined in the set of real numbers.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using various methods, including factoring, completing the square, and applying the quadratic formula. Understanding the structure of quadratic equations is crucial for applying the square root property effectively, especially when manipulating the equation to isolate the squared term.