Problem 18 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{4 x-3}=-4\)

Step-by-Step Solution

Answer

The equation has no solution in real numbers.

Step 1: Understand the Equation

The given equation is \(\sqrt{4x - 3} = -4\). We know that the square root of a real number is always non-negative. Therefore, \(-4\) is not a valid output for a square root function under real number arithmetic.

Step 2: Analyze for Real Solutions

Since the right side of the equation \(\sqrt{4x - 3} = -4\) is negative, and the square root function cannot produce a negative output, this equation has no solution in the real numbers.

Step 3: Verify the Result

By examining the properties of square root functions, we confirm that no real number solutions exist because the left-hand side, \(\sqrt{4x - 3}\), cannot equal a negative number.

Key Concepts

Square Root Function

The square root function is a crucial mathematical concept that appears often in algebra. When you see \( \sqrt{x} \), it denotes the non-negative number which, when squared, gives \( x \). This means the square root function can only yield non-negative results. If you have a square root function like \( \sqrt{y} = b \), here are some important points:

• \( y \) must be non-negative for \( \sqrt{y} \) to be defined within the real numbers.
• The function will result in \( b \geq 0 \). It cannot be negative.

These properties are essential because they establish that any square root equation where the result is negative, like \( \sqrt{4x-3} = -4 \), has some inherent issues within the realm of real numbers. Understanding this function helps identify whether a problem has realistic solutions or if it's impossible based on its initial setup.
When faced with an equation, remember to consider if a square root can indeed provide the output mentioned!

Non-negative Real Numbers

Non-negative real numbers are a fundamental part of solving equations involving square roots. They are numbers that are either greater than or equal to zero. In simpler terms, they include:

• Positive numbers (1, 2, 3, etc.)
• Zero (0)

Crucially, negative numbers are not included in non-negative real numbers. This principle is vital when working with square roots since the square root function yields only non-negative results, as discussed in the previous section.
Applying the concept of non-negative real numbers to equations can immediately hint at the nature of potential solutions. For example, an equation like \( \sqrt{4x-3} = -4 \) is impossible to solve within the reals.
Knowing this saves time and prepares you to analyze problems correctly from the start.

Equation Analysis

Equation analysis involves evaluating the components and relationships within an equation to determine possible solutions. For the equation \( \sqrt{4x-3} = -4 \), analysis begins by examining each element:

• Left-hand side: \( \sqrt{4x-3} \) represents a square root, which we've identified must equal a non-negative number.
• Right-hand side: \(-4\), provides a negative result, which conflicts with the nature of the square root.

From this analysis, it's apparent there is no real number \( x \) that can satisfy \( \sqrt{4x-3} = -4 \).
Equation analysis doesn’t stop at finding solutions; it also involves verifying them. Always check your potential answers in the context of the original equation to ensure they make sense mathematically. In cases like this, knowing the properties of components—like the square root function—can indicate from the outset if a solution is simply unseen in the real number system. Equation analysis is a handy tool to understand what you might encounter as you work through algebraic expressions.