Problem 53 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{2}+16=0 $$

Step-by-Step Solution

Answer

The equation has no real solutions; its solutions are \(x = 4i\) and \(x = -4i\).

Step 1: Write down the equation

The original equation given is \(x^2 + 16 = 0\). We need to find the solutions for \(x\).

Step 2: Isolate the quadratic term

First, subtract 16 from both sides of the equation to isolate the \(x^2\) term. This gives us \(x^2 = -16\).

Step 3: Apply the square root property

To solve for \(x\), we need to take the square root of both sides of the equation: \(x = \sqrt{-16}\). However, the square root of a negative number results in an imaginary number.

Step 4: Express the solution using imaginary numbers

Since \(\sqrt{-16}\) equals \(\pm4i\), where \(i\) is the imaginary unit (\(i^2 = -1\)), the solutions are \(x = 4i\) and \(x = -4i\).

Key Concepts

Imaginary Numbers

Imaginary numbers might sound intimidating, but they're a fascinating and useful part of mathematics. They come into play when we need to take the square root of a negative number. In the world of real numbers, taking the square root of a negative number is impossible. However, by introducing the imaginary unit denoted as "\(i\)," we can extend our number system.
The imaginary unit \(i\) is defined with a very special property: \(i^2 = -1\). This definition provides a simple yet powerful way to represent the square root of any negative number. For example, the square root of \(-16\) becomes \(\pm 4i\). The number \(4i\) is called an imaginary number because it involves the imaginary unit \(i\). This opens up a new realm of numbers that allows us to solve equations like \(x^2 + 16 = 0\), which have no solution in the real number system.

Complex Solutions

When we talk about complex solutions, what we're actually referring to are solutions that involve both real and imaginary parts. Complex numbers are in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part.
In the case of the quadratic equation \(x^2 + 16 = 0\), the solutions are \(4i\) and \(-4i\). These particular solutions are purely imaginary, meaning they have no real part (the real part \(a\) is zero). Complex solutions expand the variety of questions we can answer in mathematics.
Complex numbers are essential in many fields like engineering and physics, because they can simplify calculations and provide more complete solutions to problems that involve waves and oscillations.

Square Root Property

The square root property is a crucial technique when solving quadratic equations, especially if one side of the equation is a perfect square. This property states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\). It's a simple yet effective method that helps find solutions directly from an equation's squared term.
In our equation, \(x^2 = -16\), we apply the square root property to find \(x = \pm \sqrt{-16}\). Normally, with positive values, we would find regular numbers as roots. However, because \(-16\) is negative, the application of the square root property introduces imaginary numbers, resulting in complex solutions.
This property is particularly helpful when equations are already arranged with a squared variable, allowing quick and effective solutions without the need for factoring or using the quadratic formula.