Problem 58 - Algebra Form and Function

Question

How can you tell immediately that the equation \(x+\) \(5 \sqrt{x}=-4\) has no solutions?

Step-by-Step Solution

Answer

Question: Determine if the equation \(x + 5\sqrt{x} = -4\) has any solutions. Answer: No, the equation \(x + 5\sqrt{x} = -4\) has no solutions.

Step 1: Identify given equation

We are given an equation: \(x + 5\sqrt{x} = -4\)

Step 2: Check properties of square roots

Recall that square roots always yield non-negative values. In other words, \(\sqrt{n} \ge 0\) for any number \(n\).

Step 3: Analyze the equation

Let's analyze the left side of the equation (LHS): \(x + 5\sqrt{x}\) There are two terms in the LHS: \(x\) and \(5\sqrt{x}\). Since square roots always yield non-negative values, it's clear that \(5\sqrt{x} \ge 0\). In order for the LHS of the equation to be equal to the RHS (-4), both terms must sum to -4.

Step 4: Check if the equation can have a solution

Since \(x + 5\sqrt{x}\) must equal -4, we must require that both \(x\) and \(5\sqrt{x}\) are negative numbers, so their sum is negative. However, we established earlier that \(5\sqrt{x} \ge 0\), thus it can never be negative.

Step 5: Conclude the possibility of a solution

Based on this reasoning, the equation \(x + 5\sqrt{x} = -4\) cannot have any solutions, because the sum of \(x\) and \(5\sqrt{x}\) cannot be a negative number (-4) when one of the terms (\(5\sqrt{x}\)) is always non-negative.

Key Concepts

Square Roots

The square root function is a fundamental concept in algebra. It is important to understand that square roots are extracted from non-negative numbers. This means, for any given non-negative number \( n \), the square root \( \sqrt{n} \) is also non-negative. This is because a square root essentially asks, "What number needs to be multiplied by itself to get \( n \)?"

If you think of numbers like 4 or 16, their square roots would be 2 and 4, respectively. Both results are non-negative because \( 2 \times 2 = 4 \) and \( 4 \times 4 = 16 \). This basic property of square roots being non-negative plays a crucial role in solving or analyzing equations involving square roots, like the one in our problem where \( 5\sqrt{x} \) must be non-negative.

Non-Negative Values

Understanding non-negative values is critical in equations involving square roots. Non-negative values include all positive numbers and zero but explicitly exclude negative numbers. Mathematically, a number is non-negative if it is zero or greater than zero.

This property significantly influences equations because it restricts certain terms from having negative results. In the equation \( x + 5\sqrt{x} = -4 \), the term \( 5\sqrt{x} \) can never be negative since any square root is non-negative, and multiplying a non-negative value by a positive number (here, 5) stays non-negative.

This means no matter what value \( x \) takes, \( 5\sqrt{x} \) cannot help achieve a negative balance with any potential solution, rendering part of the equation unsolvable under these conditions.

Solution Analysis

When analyzing solutions for algebraic equations, it is essential to consider the properties of each term and the equation as a whole. Let's break down the given equation \( x + 5\sqrt{x} = -4 \).

The left-hand side contains two components: \( x \) and \( 5\sqrt{x} \). For this equation to balance and equal \(-4\) on the right-hand side, both terms would need to somehow sum up to a negative number. However, we've established that \( 5\sqrt{x} \) is non-negative, leaving us with \( x \) as potentially negative—but not negative enough to make the sum \(-4\).

Therefore, a valid analysis confirms that the equation cannot have a solution because it defies the basic properties of square roots and the nature of negative sums. This is why a thorough understanding of each term's properties and behavior in algebraic equations is so important for accurate solution analysis.