Problem 10 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ \sqrt{x}-20=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{x} - 20 = 0\) is \( x = 400\).

Step 1 - Isolating the square root

The square root can be isolated on one side of the equation by adding 20 to both sides: \( \sqrt{x}-20+20=0+20 \),which simplifies to \( \sqrt{x}=20 \).

Step 2 - Removing the square root

To remove the square root, both sides of the equation can be squared:\( (\sqrt{x})^2=(20)^2 \),which simplifies to \( x=400 \).

Step 3 - Checking for extraneous solutions

Substitute \( x = 400 \) back into the original equation to check for extraneous solutions:\( \sqrt{400}-20 \),which simplifies to 20 - 20 = 0. The answer is indeed 0, which matches the original equation. Thus, \( x = 400 \) is not an extraneous solution.

Key Concepts

Extraneous Solutions

When you solve an equation that involves a square root, you may introduce what is known as an extraneous solution. These are answers that, while they emerge from the steps of solving the equation, don't actually satisfy the original equation when substituted back in.

It's crucial to always check your solutions by plugging them back into the original equation, because when you square both sides of an equation (which is a common step in solving square root equations), you might include a solution that doesn't make mathematical sense in the context of the problem. This can occur because squaring is a non-reversible process, which means squaring both sides can sometimes lead to a loss of information about the sign (positive or negative) of the original expressions.

Isolating the Square Root

The strategy of isolating the square root on one side of an equation is pivotal for clear and efficient problem solving. This step involves manipulating the equation so that the square root term stands alone.

To achieve this, you'll perform inverse operations, such as addition or subtraction, to cancel out any other terms that are with the square root. This simplifies the equation, making it easier to deal with, and sets the stage for the next step, which is to eliminate the square root entirely.

Removing the Square Root

Removing the square root from an equation is a necessary step in finding the solution to square root equations. This is commonly done by squaring both sides of the equation, as the square of a square root will give you the original expression inside the square root.

However, it's important to handle this step with care, because while it clears away the square root, it can also introduce the possibility of extraneous solutions, as previously discussed. The process of squaring can simplify the equation's form, but it's a sensitive operation that has implications for the solution's validity.

Equation Solving Steps

An organized series of equation solving steps can streamline the path to finding the solution to square root equations. The typical steps include isolating the square root, squaring both sides to remove the square root, simplifying the resulting equation, solving for the variable, and then checking for extraneous solutions.

Each step serves a specific purpose and must be performed with diligence to ensure you're not led astray by potential pitfalls, such as the introduction of extraneous solutions. Clear and methodical steps guarantee that every possible solution is accounted for and the final answer truly satisfies the original equation.