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 is in the form \( (x - a)^2 = k \), where \(a = 3\) and \(k = -5\). The square root property states that if \( (x - a)^2 = k \), then \( x - a = \pm \sqrt{k} \).
2. Apply the square root property to the equation: write \( x - 3 = \pm \sqrt{-5} \).
3. Since the square root of a negative number involves imaginary numbers, express \( \sqrt{-5} \) as \( \sqrt{5}i \), where \(i\) is the imaginary unit with the property \( i^2 = -1 \). So, rewrite the equation as \( x - 3 = \pm \sqrt{5}i \).
4. Isolate \(x\) by adding 3 to both sides: \( x = 3 \pm \sqrt{5}i \).
5. Write the final solution as two complex numbers: \( x = 3 + \sqrt{5}i \) and \( x = 3 - \sqrt{5}i \). These are the solutions to the equation using the square root property.

Key Concepts

Square Root Property

The square root property states that if an equation is in the form (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 not real but complex. The imaginary unit i is defined as √(-1), enabling the expression of roots of negative numbers as multiples of i, such as √(-5) = i√5.

Isolating the Variable

Before applying the square root property, the equation must be manipulated to isolate the squared term on one side. This involves algebraic steps like adding or subtracting terms to simplify the equation to the form (x - a)^2 = b.