Problem 57 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ x^{2}+5=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \pm \sqrt{5}i \).

Step 1: Isolate the Quadratic Term

Our goal is to isolate the equation to find the solution for \( x \). Start by isolating the quadratic term \( x^2 \) by subtracting 5 from both sides of the equation:\[x^2 = -5\]

Step 2: Apply the Square Root Property

To solve for \( x \), take the square root of both sides. Remember that the square root of a negative number involves imaginary numbers:\[x = \pm \sqrt{-5}\]

Step 3: Simplify the Square Root

Simplify the square root by recognizing that \( \sqrt{-1} = i \), where \( i \) is the imaginary unit:\[x = \pm \sqrt{5}i\]

Step 4: Write the Answers in Standard Form

Write the solutions in the standard complex number form \( a + bi \):\[x = 0 + \sqrt{5}i \quad \text{and} \quad x = 0 - \sqrt{5}i\]

Key Concepts

Complex Numbers

Complex numbers are an extension of the familiar number system you are used to, like integers and real numbers. They have the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. What makes complex numbers so powerful is their ability to express solutions to equations that have no real solutions, like the square root of a negative number.

• Real Part: The \(a\) in \(a + bi\).
• Imaginary Part: The \(b\) in \(a + bi\), multiplied by \(i\).

In our exercise, the solutions are complex numbers because they contain an imaginary part. There is no real solution to \(x^2 + 5 = 0\) as real numbers do not satisfy the equation \(x^2 = -5\). Instead, the solutions are \(0 + \sqrt{5}i\) and \(0 - \sqrt{5}i\), which are complex because they involve \(i\), the imaginary part.

By understanding complex numbers, we open new possibilities in solving equations that were previously unsolvable with just real numbers.

Imaginary Number

The concept of imaginary numbers can be a real brain-buster at first. The term 'imaginary' might suggest that they don't "exist"—but that's not true. They are just as real in math as any other concept and are extremely useful.

The imaginary unit \(i\) is defined as \(\sqrt{-1}\). So, when you need to take the square root of a negative number, you express it using \(i\). For instance, \(\sqrt{-5}\) can be re-expressed as \(\sqrt{5}i\). This is used in our problem to find solutions to \(x^2 + 5 = 0\).

Here are some properties of \(i\):

• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

These properties enable mathematicians to handle powers of imaginary numbers easily, and are especially handy when simplifying expressions.

Square Root Property

The square root property is a mathematical method used to solve equations of the form \(x^2 = c\). This property states that if \(x^2 = c\), then \(x = \pm \sqrt{c}\). It's important to remember the "plus-minus" (\(\pm\)) when solving these equations, because both positive and negative roots can be solutions.

In our exercise, we isolate \(x^2\) on one side of the equation, resulting in \(x^2 = -5\). Since \(-5\) is negative, applying the square root property would usually not give us real numbers. Instead, we use imaginary numbers to express \(\pm \sqrt{5}i\).

Using the square root property with complex numbers:

• Recognize that you will have imaginary solutions if \(c\) is negative.
• You can rewrite \(\sqrt{-c}\) as \(\sqrt{c}i\).

Understanding the square root property is essential for dealing with quadratic equations, especially when they don't have real number solutions.