Problem 74 - Algebra 2

Question

For what values of \(x\) is \(\sqrt{x} \cdot \sqrt{-x}\) a real number? Explain.

Step-by-Step Solution

Answer

The only value for \(x\) that makes \(\sqrt{x} \cdot \sqrt{-x}\) a real number is when \(x=0\).

Step 1: Analyzing the equation

The first step is to understand that the square root of a negative number is imaginary. This comes from the definition of the imaginary unit 'i', which is \(\sqrt{-1}\). Therefore, \(\sqrt{-x}\) would be equal to \(\sqrt{x} \cdot i\) since the square root of a negative can be understood as the square root of the corresponding positive number multiplied by 'i'.

Step 2: Substitution

Substitute \(\sqrt{-x}\) with \(\sqrt{x} \cdot i\). So, now the equation becomes \(\sqrt{x} \cdot (\sqrt{x} \cdot i)\). This simplifies to \(x \cdot i\).

Step 3: When is this product a real number?

The product \(x \cdot i\) can be a real number only if \(x\) is equal to zero. For any other real number, the product becomes an imaginary number. Therefore, the only value of \(x\) that makes \(\sqrt{x} \cdot \sqrt{-x}\) a real number is \(x=0\).

Key Concepts

Imaginary Numbers

Imaginary numbers come into play when dealing with the square roots of negative numbers, which naturally do not exist on the real number line. The imaginary unit, denoted by 'i', is defined as the square root of -1, i.e., \(\sqrt{-1} = i\). This fundamental definition helps us work with negative square roots.

By understanding this, you're equipped to handle expressions like \(\sqrt{-x}\). This is equivalent to \(\sqrt{x} \cdot i\), where \(x\) is a positive number. Here, '\(x\)' becomes the non-negative component while '\(i\)' handles the negative aspect, transforming what would otherwise be undefined into a crisp, usable mathematical expression.

Such manipulation allows for consistent and logical interpretations of square roots in mathematical expressions, enabling solutions where none were apparent in the realm of real numbers.

Real Numbers

Real numbers encompass all rational and irrational numbers, serving as the vast universe where most everyday arithmetic operates. They include numbers like integers, fractions, and even roots of positive numbers. However, when it comes to the square roots of negative numbers, real numbers fall short, as they simply can't be computed within the real realm.

The problem posed, \(\sqrt{x} \cdot \sqrt{-x}\), tests the boundaries of real numbers. Normally, a square root product is real if both components remain in the realm of real numbers.

In the equation, once \(\sqrt{-x}\) becomes \(\sqrt{x} \cdot i\), the result \(x \cdot i\) moves away from being real because of the presence of 'i'. The exception is when \(x = 0\), where the product zeroes out the imaginary component, returning us safely to the real number \(0\). Thus, \(x = 0\) is the sole value satisfying the condition of realness in this context.

Identity Properties

Identity properties are fundamental concepts that make complex arithmetic much simpler. They can help us understand and manipulate mathematical expressions with confidence. In the realm of both real and imaginary numbers, these properties ensure consistency.

There are two main identity properties in arithmetic:

• **Additive Identity**: Adding 0 to any number doesn't change its value. For example, \(a + 0 = a\).
• **Multiplicative Identity**: Multiplying any number by 1 doesn't alter the number. For example, \(a \times 1 = a\).

These properties ensure we can safely perform operations without changing the underlying values.

In the context of solving \(\sqrt{x} \cdot \sqrt{-x}\), the zero product property (a type of identity property) becomes crucial. When \(x = 0\), the product \(x \cdot i\) simplifies because any number multiplied by 0 results in 0. Thus achieving the condition for a real number without invoking imaginary components makes it clear why zero is the crucial solution.