Problem 15 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$\sqrt{x}-9=0$$

Step-by-Step Solution

Answer

The solution to the equation \( \sqrt{x} - 9 = 0 \) is \( x = 81 \)

Step 1: Isolate the square root

To solve the equation, start by isolating the square root on one side of the equation which is already done and the equation becomes \( \sqrt{x} = 9 \).

Step 2: Get rid of square root

To get rid of the square root, square both sides of the equation. This results in \( x = 9^2 \) or \( x = 81 \).

Step 3: Check for extraneous solutions

Now, to check for extraneous solutions, substitute the solution back into the original equation. If it satisfies the original equation, then it's a valid solution. In this case, it's \( \sqrt{81} - 9 = 0 \) which becomes \( 9 - 9 = 0 \). So, \( x = 81 \) is a valid solution, and there are no extraneous solutions.

Key Concepts

Extraneous Solutions

An important aspect when solving equations involving square roots is the consideration of extraneous solutions. These are results that emerge from the process of solving the equation but do not hold true when plugged back into the original equation. Extra caution is necessary because when we perform operations like squaring both sides of an equation, we're potentially introducing solutions that may not be true to the initial scenario.

For instance, if you solve an equation and get a negative number when the original equation involves a square root, you know there's a mistake, because square roots by definition yield non-negative results. To avoid accepting false solutions, re-substitute any derived solutions into the original equation to validate them. In our example, where the derived solution was 81, it was successfully validated, hence, it was not extraneous.

Isolating the Square Root

When tackling square root equations, the priority is to isolate the square root term on one side of the equation. This helps in simplifying the process of finding solutions.

Isolating the square root involves moving all other terms to the opposite side of the equation using basic algebraic operations like addition or subtraction. Once the square root term stands alone, it can be eliminated by raising both sides of the equation to the power of two, which effectively removes the square root. However, be mindful that this must be done cautiously as it can potentially introduce extraneous solutions, which we've discussed previously.

Solving Radical Equations

Solving radical equations, which are equations containing a square root or other root expressions, requires a systematic approach. The goal is to free the variable from inside the radical. After isolating the square root, you eliminate the radical by applying an operation that undoes the radical; squaring in the case of square roots. This should give you an algebraic equation that can often be solved with simple algebraic methods.

Always remember to perform a check at the end by substituting the solution back into the original equation to ensure its validity, thereby identifying or ruling out extraneous solutions. While working through these steps, bear in mind the mathematical properties of radicals and the implications of each operation performed on the equation.