Problem 25 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ (x+3)^{2}=-16 $$

Step-by-Step Solution

Answer

The solutions of the given equation are \(x = 4i -3\) or \(x = -4i -3\).

Step 1: Apply the square root property

The first step is to apply the square root property to both sides of the equation to get rid of the square in the left-hand side. However, since the right-hand side of the equation is -16, the result of square root will be an imaginary number. The equation becomes: \(x+3 = i\sqrt{16}\) or \(x+3 = -i\sqrt{16}\).

Step 2: Simplify the equation

In the second step, we simplify the square root in both equations to get: \(x+3= 4i\) or \(x+3= -4i\).

Step 3: Solve for x

The final step is to subtract 3 from both sides of each equation to solve for 'x'. This yields: \(x = 4i -3\) or \(x = -4i -3\).

Key Concepts

Imaginary Numbers

Imaginary numbers are an intriguing and essential concept in mathematics, specifically in complex number theory. They are numbers that give a negative result when squared, something that is impossible under the rules of real numbers. For instance, consider the equation \(x^2 = -1\). The square of a real number can never be negative, so to handle such scenarios, mathematicians use the imaginary unit, denoted as \(i\), where \(i^2 = -1\).

Now, how does this apply to our original exercise, where we encountered \(x+3)^2 = -16\)? After applying the square root property, we face the square root of a negative number. This is where imaginary numbers are utilized; the square root of -16 becomes \(4i\), as \(i\) represents the square root of -1. This permits the continuation of solving the equation while staying within the bounds of mathematical rules and integrity.

Solving Quadratic Equations

Quadratic equations are a fundamental part of algebra, with the general form being \(ax^2 + bx + c = 0\). Solving these can be approached in various ways, including factoring, completing the square, using the quadratic formula, and the method in our exercise, utilizing the square root property.

The square root property states that if \(x^2 = d\), then \(x = \pm\sqrt{d}\). In our original exercise, to isolate \(x\), we must first eliminate the squaring operation, which is done by taking the square root of both sides of the equation. However, the presence of a negative number on the right side leads us to involve imaginary numbers. After reaching \(x+3 = \pm4i\), the last step is straightforward algebra: subtract 3 from both sides to solve for \(x\). This technique is especially powerful because it simplifies the equation to a degree where the solution becomes obvious.

Complex Numbers

Complex numbers represent a broadened concept of what numbers can be, compared to real numbers. A complex number consists of a real part and an imaginary part, and is typically written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In our equation, solutions \(x = 4i - 3\) and \(x = -4i - 3\) are both complex numbers.

It's important to understand that complex numbers are not 'imaginary' in the sense that they don't exist; rather, they are 'imaginary' in that they exist in a dimension of mathematics that extends beyond the one-dimensional line of real numbers. Complex numbers are a fundamental part of modern mathematics and physics, as they provide solutions to equations that do not have real number answers, making them indispensable in many domains of science and engineering.