Problem 32 - Elementary and Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{3 x}+5=2 $$

Step-by-Step Solution

Answer

There are no real solutions to the equation.

Step 1: Isolate the square root

We begin by isolating the square root term. Subtract 5 from both sides of the equation: \[ \sqrt{3x} + 5 - 5 = 2 - 5 \] This simplifies to: \[ \sqrt{3x} = -3 \]

Step 2: Consider the properties of square roots

The square root of a real number, \(\sqrt{3x}\), is always non-negative. Therefore, \(\sqrt{3x} = -3\) has no real solutions since a square root cannot equal a negative number.

Step 3: Conclude no viable solutions

Being unable to find a valid \(x\) that satisfies the condition \(\sqrt{3x} = -3\), we determine that there are no real solutions to the equation.

Key Concepts

Understanding Square Root Properties

Square roots are fascinating because they involve finding a number which, when multiplied by itself, gives the original number under the radical sign. A key property of square roots is that they are never negative. This means whenever you see \( \sqrt{a} \), the value is always non-negative, assuming \( a \) is a non-negative real number. In our problem, we have \( \sqrt{3x} \), and according to square root properties, it cannot be less than zero. So immediately, if you come across an equation like \( \sqrt{3x} = -3 \), you know something isn't right, since a square root can't be negative. Understanding these properties helps decide if an equation might have no solution. If an equation insists that a square root equals a negative number, that might direct us to check our steps or reassess the validity of the solutions involved.

Isolating Terms in Equations

When solving an equation, isolating the variable term is a crucial step. It's like peeling away the outer layers of an onion to get to the center. Here, isolating means manipulating the equation until the term you are interested in is all on its own. In our given problem, we first had \( \sqrt{3x} + 5 = 2 \). To isolate \( \sqrt{3x} \), we subtract 5 from both sides of the equation, leading to \( \sqrt{3x} = -3 \).However, while isolating terms is usually a step toward finding solutions, sometimes it leads to a realization about the nature of potential solutions. In this problem, it highlights that there is no valid value of \( x \) that would work because the square root must be non-negative. Being able to efficiently isolate terms lets you quickly figure out how to proceed further.

Understanding Extraneous Solutions

Extraneous solutions are unintended results that arise when solving equations, often due to the algebraic manipulation process. They are solutions that appear technically correct but don't satisfy the original equation. In this context, when we rearrange and manipulate the square root equation to solve for \( x \), it's crucial to double-check whether the "solutions" actually make sense in the original context. An extraneous solution might come up due to steps like squaring both sides in different types of problems, but in our case, we quickly identify there are no real solutions because of the property that square roots must be non-negative. Leaving no room for an extraneous solution here, we conclude the problem by realizing that no \( x \) can ever satisfy \( \sqrt{3x} = -3 \). This helps us avoid mistaken correctness and save time by verifying the legitimacy of the results we work out.