Problem 42 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ \sqrt{x}+5=1 $$

Step-by-Step Solution

Answer

The equation has no solution as the square root of a number cannot equal a negative number.

Step 1: Isolate the square root term

The first step is to isolate the expression containing the square root. This can be done by subtracting 5 from both sides of the equation. Doing so, we get:\[ \sqrt{x} + 5 - 5 = 1 - 5 \]which simplifies to\[ \sqrt{x} = -4 \]

Step 2: Identify extraneous solutions

The square root of a number is always nonnegative (i.e., positive or zero). We see here that the square root of \(x\) supposedly equals a negative number. This is impossible, meaning that the provided equation has no solution.

Key Concepts

Square Root Equations: Nature and Solutions

Square root equations involve terms under a square root symbol, commonly solving for an unknown that makes these terms true. When dealing with square root equations, the challenge is often finding the unknown value that satisfies the equation.

To solve these equations:

• First, isolate the square root term on one side if it's not already isolated. This makes it easier to square both sides and eliminate the square root.
• Squaring both sides of the equation can help remove the square root, but be careful, as this operation might introduce extraneous solutions (solutions that don't actually satisfy the original equation).

In our original problem, we started with the equation \( \sqrt{x} + 5 = 1 \). The goal was to isolate the square root \( \sqrt{x} \), and we see the importance of careful handling to avoid erroneous conclusions.

Isolating Expressions: Key Techniques

Isolating expressions in algebra is crucial when solving equations. The idea is to have a single instance of the unknown variable or expression on one side of the equation.

Here's how to effectively isolate an expression:

• Perform inverse operations to both sides of the equation. For instance, if something is added to the expression, you subtract it from both sides, and vice versa.
• Be methodical, performing one operation at a time to simplify the equation.

In the given exercise, isolating \( \sqrt{x} \) involved subtracting 5 from both sides, simplifying the equation to \( \sqrt{x} = -4 \). This step is crucial, revealing both the nature of the equation and potential impossibilities when looking for solutions.

Nonnegative Numbers: Understanding Their Role

A pivotal concept in this exercise is understanding nonnegative numbers. A nonnegative number is any number that is either positive or zero. When dealing with square roots, the results are always nonnegative since squaring any real number yields a positive result or zero.

Here’s why it matters:

• Square root functions only output nonnegative results in conventional arithmetic, which is important when determining the viability of solutions in square root equations.
• If isolating the square root yields a negative number, as in our exercise, it signals an impossible scenario under real numbers.

In the provided problem, once we reached \( \sqrt{x} = -4 \), we identified this as a key indicator that the equation has no real solution, thereby avoiding extraneous solutions.