Problem 48 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ \sqrt{x}-5=20 $$

Step-by-Step Solution

Answer

The solution to \( \sqrt{x}-5=20 \) is \( x = 625 \).

Step 1: Isolate the square root expression

First, add 5 to both sides of the equation to isolate the square root expression on one side. This yields the equation \( \sqrt{x} = 25 \)

Step 2: Eliminate the square root

To get the x by itself, square both sides of the equation to remove the square root. Squaring \( \sqrt{x} = 25 \) gives \( x = 625 \)

Step 3: Check for potential extraneous solutions

Substitute \( x = 625 \) into the original equation to check if it is a valid solution. \( \sqrt{625}-5=20? \) The left side of the equation is 25 which is equal to the right side of the equation, so \( x = 625 \) is a valid solution.

Key Concepts

Understanding Extraneous Solutions

When solving radical equations, the process of squaring both sides can sometimes introduce extraneous solutions. An extraneous solution is a result that comes from the algebraic manipulation, but does not satisfy the original equation. This happens because when both sides of an equation are squared, it might create new solutions that are not valid for the original problem. So, although it might appear like a solution mathematically, it may not actually fulfill the equation it was meant to solve. To ensure accuracy, it's crucial to verify every potential solution by substituting it back into the original equation. This check helps confirm whether the solution works or not, weeding out any extraneous answers.

Importance of Checking Solutions

Checking solutions plays a vital role in ensuring that the answers you find truly satisfy the equation you started with. This step cannot be overlooked, especially when dealing with radical equations. After solving an equation, it's important to substitute the potential solution back into the original equation to see if it holds true.

• Start by substituting the solution into the original equation.
• Perform all necessary calculations on both sides of the equation.
• Check if the left side equals the right side.

If they are equal, then the solution is valid. If they are not, then you have to discard that solution as it is extraneous. This ensures that any solution you conclude is accurate and representative of the problem given.

Isolating Square Roots

Isolating the square root is often the initial step in solving an equation involving radicals. The goal here is to have the square root on one side of the equals sign with everything else on the other.

• Look at the equation and identify the square root term.
• Use algebraic operations to move other terms away from the square root term, usually by addition or subtraction.
• Ensure that nothing but the square root remains on one side of the equation.

For the given equation \( \sqrt{x} - 5 = 20 \), the isolation process involves moving the \(-5\) to the other side by adding 5 to both sides. This simplifies the task of removing the square root since it makes it the focal point of the equation. Once isolated, you can move on to more straightforwardly solve for the variable by eliminating the square root, often through squaring both sides of the equation.