Problem 94 - Elementary and Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{3 x}+10=1 $$

Step-by-Step Solution

Answer

No real solutions exist because the square root cannot be negative.

Step 1: Isolate the Square Root

Start by isolating the square root term on one side. Subtract 10 from both sides of the equation: \( \sqrt{3x} = 1 - 10 \), which simplifies to \( \sqrt{3x} = -9 \).

Step 2: Analyze the Equation

Consider the isolated equation \( \sqrt{3x} = -9 \). Square roots result in non-negative numbers; therefore, no real number squared will give a negative result.

Step 3: Identify Extraneous Solutions

Since \( \sqrt{3x} = -9 \) is impossible for any real number \( x \), there are no solutions to this equation.

Key Concepts

Understanding Square Roots

Square roots are mathematical functions that help us determine the original number that was squared. When you see the square root symbol (\( \sqrt{} \)), it means we are looking for a number which, when multiplied by itself, gives us the original number inside the square root.

For example, \( \sqrt{16} \) is 4 because \( 4 \times 4 = 16 \). Importantly, square roots always yield non-negative results. This is crucial to solving equations containing square roots, as any negative result would not have a real number solution.

• Square root can be expressed as \( \sqrt{x} \).
• Results in a non-negative number.

When isolating a square root to solve an equation, be sure to consider what numbers, if any, can realistically be produced. If a square root equals a negative number, as seen in our specific problem \( \sqrt{3x} = -9 \), it indicates that no solution exists in the realm of real numbers because we cannot square a real number to get a negative outcome.

Identifying Extraneous Solutions

Extraneous solutions are outcomes that appear during the process of solving an equation but are not valid for the original equation. These often occur when operations like squaring both sides are used. Squaring can introduce solutions not initially present in the equation.

To identify extraneous solutions, it's essential to go back and substitute the solution back into the original equation. If the solution doesn't satisfy the original equation, it is considered extraneous.

• Caused by manipulating the equation.
• Do not satisfy the original equation.

In our exercise, after isolating the square root \( \sqrt{3x} = -9 \), it's evident right away there are no solutions, as negative results from square roots are impossible for real numbers. Hence, there are no checks needed for extraneous solutions, as no real solution exists.

Exploring Real Numbers

In mathematics, real numbers include all the numbers you might encounter in everyday life, encompassing both rational and irrational numbers. They include positive numbers, negative numbers, and zero, though they exclude imaginary numbers.

• Real numbers include rationals like \( \frac{1}{2} \) and \( -3 \), as well as irrationals like \( \sqrt{2} \).
• Cannot be negative when involving square roots in equations.

When solving equations like \( \sqrt{3x} = -9 \), you're operating within the set of real numbers. Any assertion that a square root equals a negative number is a signal that you're venturing out of real number territory, as no real number multiplied by itself is negative.