Problem 50 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ 3 \sqrt{x}=-21 $$

Step-by-Step Solution

Answer

The equation has no real solutions as the obtained solution \(x = 49\) is extraneous.

Step 1: Isolate the Square Root

The square root is already isolated on one side of the equation: \(3 \sqrt{x} = -21\).

Step 2: Cancel Out the Coefficients

To get rid of the coefficient of the square root, divide both sides of the equation by 3. The equation becomes \(\sqrt{x} = -7\).

Step 3: Squaring Both Sides of the Equation

Square both sides of the equation to eliminate the square root: \( (\sqrt{x})^2 = (-7)^2 \), simplifying this leads to: \(x = 49\).

Step 4: Checking for Extraneous Solutions

Substitute \(x = 49\) into the original equation to check for extraneity. This gives us \(3 \sqrt{49} = 3*7 = 21 \), which isn't equal to -21. Hence, \(x = 49\) is an extraneous solution.

Key Concepts

Extraneous Solutions

Sometimes, when you solve equations involving square roots, you might obtain what are called extraneous solutions. These are solutions that arise from the process of solving the equation, but don’t satisfy the original equation. This typically happens when certain mathematical operations like squaring both sides alter the nature of the problem.

Such operations can introduce solutions that aren't valid when plugged back into the initial equation. Here’s how you can manage them:

• Always verify your potential solutions by substituting them back into the original equation.
• Be cautious with operations that might expand the solution set, like squaring.

These steps ensure you only accept the solutions that truly satisfy the equation.

Isolate Square Roots

Isolating square roots is an essential step when solving equations involving them. The goal is to have the square root term by itself on one side of the equation. If the square root is not yet isolated, rearrange the equation by performing addition, subtraction, or dividing both sides as needed.

In our exercise, the equation starts as \(3 \sqrt{x} = -21\). The square root \(\sqrt{x} \) is multiplied by a coefficient. To isolate it, you would divide both sides of the equation by 3, leading to \(\sqrt{x} = -7\). At this stage, the square root is isolated and easy to work with in the next steps.

Squaring Both Sides

When you square both sides of an equation, it eliminates the square root, allowing you to solve more easily.

After isolating the square root, we have \(\sqrt{x} = -7\). Squaring both sides will result in \((\sqrt{x})^2 = (-7)^2\). This simplifies to \(x = 49\).

• This step transforms the square root into a solvable form.

However, always remember to check if the solution is extraneous. Sometimes squaring can introduce invalid solutions that need to be verified using the original equation, as it might lead to false positives.