Problem 31 - Elementary and Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{5 x}+10=8 $$

Step-by-Step Solution

Answer

The equation has no valid solutions; \(x = \frac{4}{5}\) is extraneous.

Step 1: Isolate the Square Root

Start by isolating the square root on one side of the equation.\[ \sqrt{5x} + 10 = 8 \]. Subtract 10 from both sides to get \[ \sqrt{5x} = -2 \].

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root.\[ (\sqrt{5x})^2 = (-2)^2 \]. This simplifies to \( 5x = 4 \).

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by 5.\[ x = \frac{4}{5} \].

Step 4: Check for Extraneous Solutions

Substitute \(x = \frac{4}{5}\) back into the original equation to verify if it's correct. Calculate \( \sqrt{5 \times \frac{4}{5}} + 10 = 8 \). This simplifies to \(\sqrt{4} + 10 = 8 \), or \(2 + 10 = 8 \). This statement is false, so \(x = \frac{4}{5} \) is an extraneous solution.

Key Concepts

Square Roots

Square roots are a fundamental concept in algebra and mathematics in general. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, since 4 times 4 equals 16. In algebra, square roots are often involved in equations that require solving for unknown variables, such as the equation \( \sqrt{5x} + 10 = 8 \).To solve equations with square roots, the first step is to isolate the square root term on one side. This helps to focus on removing the square root by squaring both sides of the equation. Doing so will eliminate the square root and often simplify the equation significantly. However, one must be cautious as squaring both sides can introduce extraneous solutions, which are not actual solutions to the original equation.

Extraneous Solutions

Extraneous solutions are solutions that arise when solving certain types of equations, but do not satisfy the original equation. They often occur in the process of solving equations involving square roots or other logical operations like absolute values.When solving for a variable by squaring both sides of an equation, as seen in \( \sqrt{5x} = -2 \), this step may introduce solutions that do not exist in the realm of basic arithmetic (e.g., a real square root cannot be negative). The critical part about handling extraneous solutions is to always verify your results by plugging them back into the original equation. If they do not satisfy the initial equation, they should be considered extraneous. This ensures the integrity and correctness of your final answer.

Step-by-Step Solution

A step-by-step solution method is incredibly helpful in tackling algebraic problems, especially for students learning to solve their first equations. Breaking down the process into manageable steps allows for clear understanding and follow-up at every phase of problem-solving. Let's take the original equation \( \sqrt{5x} + 10 = 8 \) as an example:

• **Step 1**: Isolate the square root. We get \( \sqrt{5x} = -2 \) by subtracting 10 from both sides.
• **Step 2**: Square both sides to remove the square root, resulting in \( 5x = 4 \).
• **Step 3**: Solve for \(x\) by dividing both sides by 5, giving \( x = \frac{4}{5} \).
• **Step 4**: Verify the solution by substituting back into the original equation, finding it does not work, thus identifying it as extraneous.

A structured solution like this not only aids in finding solutions but also in understanding the intricacies of algebraic operations and recognizing potential pitfalls, like extraneous solutions.