Problem 75 - Beginning and Intermediate Algebra

Question

Solve each equation. (All solutions are nonreal complex numbers.) $$ x^{2}=-26 $$

Step-by-Step Solution

Answer

\(x = \pm \sqrt{26} i\)

Step 1 - Understand the problem

The given equation is a quadratic equation with the form \(x^2 = -26\). The goal is to find the nonreal complex solutions for \(x\).

Step 2 - Isolate the variable

The equation \(x^2 = -26\) shows that \(x^2\) equals a negative number, which means that the solutions will be complex numbers. To isolate \(x\), take the square root of both sides of the equation.

Step 3 - Apply the square root

Applying the square root to both sides gives: \(x = \pm\sqrt{-26}\). Remember that the square root of a negative number involves the imaginary unit \(i\). So, \(\sqrt{-26} = \sqrt{26} \cdot i\).

Step 4 - Simplify the complex expression

Simplifying the expression, we get: \(x = \pm \sqrt{26} i\).

Step 5 - State the solutions

The solutions to the equation \(x^2 = -26\) are: \(x = \pm \sqrt{26} i\).

Key Concepts

Nonreal Complex Numbers

In the realm of mathematics, numbers aren't just confined to real numbers. When we venture into the world of complex numbers, we encounter nonreal complex numbers. These numbers include a real part and an imaginary part.

A complex number is expressed in the form: \(a + bi\), where:

• \(a\) is the real part.
• \(bi\) is the imaginary part.

Nonreal complex numbers are those where the imaginary part (\(bi\)) is not zero. For instance, \(3 + 4i\) and \(-5 + 2i\) are nonreal because they contain an imaginary component.
Understanding nonreal complex numbers is essential when solving quadratic equations that yield negative results under the square root. As with the given equation \(x^2 = -26\), the solutions are nonreal complex numbers because the square of no real number gives a negative product.

Imaginary Unit

To deal with the square roots of negative numbers, mathematicians introduced the concept of the imaginary unit, denoted as \(i\). The imaginary unit \(i\) is defined by the fundamental property:

\(i^2 = -1\)

This definition allows us to express the square roots of negative numbers in terms of \(i\). For example, \(\sqrt{-1} = i\). This conversion is very useful for solving equations like \(x^2 = -26\), where taking the square root of both sides involves an imaginary component.

Applying the square root to both sides of \(x^2 = -26\) gives us \(x = \pm\sqrt{-26}\). We can break down \(\sqrt{-26}\) using \(i\):

\[\sqrt{-26} = \sqrt{26} \cdot i\]

So, our solutions become \(x = \pm \sqrt{26} i\), solidifying our understanding and usage of the imaginary unit \(i\).

Square Root of Negative Numbers

One of the intriguing aspects of mathematics arises when we work with the square root of negative numbers. Traditional square root functions, when applied to negative numbers, don't produce real results. This is where complex numbers and the imaginary unit step in.

When we encounter a negative number beneath the square root sign, we use the imaginary unit \(i\) to transform it into a comprehensible form. For example:

\[\sqrt{-a} = \sqrt{a} \cdot i\]

Here, \(a\) represents a positive real number.

Let's apply this to our exercise: The equation \(x^2 = -26\) required us to find \(\sqrt{-26}\). Following the rule, we rewrite it as:

\[\sqrt{-26} = \sqrt{26} \cdot i\]

This transformation allows us to find the solutions in the form of complex numbers. Thus, the equation \(x^2 = -26\) has solutions \(x = \pm \sqrt{26} i\). This approach makes it possible to handle negative values under the square root, leading to nonreal complex solutions.