Problem 4 - Beginning and Intermediate Algebra

Question

The equation \(x^{2}=-9\) has solutions that (are/are not) real numbers. These solutions involve the imaginary unit \(i,\) which is defined as \(i=\) ________ Thus, \(i^{2}=\) ____________ For any positive real number \(a\), we have \(\sqrt{-a}=\) __________.

Step-by-Step Solution

Answer

The solutions are not real numbers. The imaginary unit \(i\) is defined as \sqrt{-1}\. Therefore, \(i^{2} = -1\) and \( \sqrt{-a} = \sqrt{a}i\).

Step 1: Identify the type of solutions

The equation given is \(x^{2} = -9\). Since the right-hand side of the equation is negative, the solutions involve imaginary numbers.

Step 2: Introduce the imaginary unit

The imaginary unit \(i\) is defined as the square root of \(-1\). Thus, \(i = \sqrt{-1}\).

Step 3: State the square of the imaginary unit

If \(i = \sqrt{-1}\), then \ i^{2} = -1\.

Step 4: Express the square root of negative numbers

For any positive real number \(a\), the square root of \( -a\) can be written as: \( \sqrt{-a} = \sqrt{a}i\).

Key Concepts

imaginary unit

When dealing with equations that have negative numbers inside a square root, we use the imaginary unit, denoted as \(i\). The imaginary unit is defined as the square root of \(-1\), so \(i = \sqrt{-1}\). This allows us to handle square roots of negative numbers in a consistent way.

For example, in the equation \(x^{2} = -9\), we need to find the square root of \(-9\). By introducing the imaginary unit, we express it as \(x = \pm 3i\), since \(3i\) squared gives \(-9\).

Always remember:

• The imaginary unit \(i\) is crucial for working with negative numbers inside square roots.
• \(i\) is defined as \(\sqrt{-1}\).
• \(i^{2}\) equals \(-1\).

Understanding the imaginary unit simplifies working with complex and imaginary numbers.

square root

The square root is a mathematical operation that asks the question: which number, when multiplied by itself, gives the original number? For example, the square root of 9 is 3, because \(3 \times 3 = 9\). However, when dealing with negative numbers, things get tricky.

Normally, the square root of a negative number is not a real number because no real number squared gives a negative result. This is where the imaginary unit \(i\) comes in. For any positive real number \(a\), the square root of \(-a\) can be expressed as \(\sqrt{-a} = \sqrt{a}i\).

By using the imaginary unit:

• We can transform the square root of a negative number into a product of a real number and \(i\).
• For instance, \(\sqrt{-9} = \sqrt{9}i = 3i\).

This makes it easier to work with and understand equations involving negative square roots.

equations with negative numbers

Equations involving negative numbers, particularly under a square root, lead us into the realm of imaginary numbers. Consider the equation \(x^{2} = -9\). At first glance, finding \(x\) might seem impossible because no real number squared gives a negative result.

However, by introducing the concept of the imaginary unit \(i\), we handle these equations effectively. We recognize that the solutions are not real numbers but involve \(i\).

Here's how it works step-by-step:

• Given \(x^{2} = -9\), we rewrite it as \(x = \pm \sqrt{-9}\).
• Using the imaginary unit, we express \(\sqrt{-9}\) as \(\sqrt{9}i\).
• Thus, \(x = \pm 3i\).

This method simplifies solving and understanding such equations, showing that math has tools to handle even the most challenging problems.