Problem 5 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ 14=\sqrt{x} $$

Step-by-Step Solution

Answer

The solution to the equation \(14=\sqrt{x}\) is \(x = 196\).

Step 1: Isolate the square root

Start by isolating the square root on one side of the equation. In this case \( \sqrt{x} \) is already isolated. So, the equation remains: \( 14 = \sqrt{x} \)

Step 2: Square both sides

In order to remove the square root, square both sides of the equation: \( (14)^2 = (\sqrt{x})^2 \). This simplifies to: \( 196 = x \)

Step 3: Check for extraneous solutions

It's critical to verify no extraneous solutions have been introduced during the calculation process. To check, plug \$x = 196\$ back into the original equation \(14 = \sqrt{x}\), which simplifies to \( 14 = \sqrt{196}\). Since \( 14 = 14 \), we see that \$x = 196\$ is indeed the valid solution to the equation.

Key Concepts

Understanding Extraneous Solutions

When solving square root equations, sometimes extra solutions appear, called extraneous solutions. These happen because of the steps we use to solve equations, like squaring both sides. Squaring can introduce solutions that don't work in the original formula. Ensuring these solutions are identified is important because they can lead us to incorrect conclusions if left unchecked. Extraneous solutions often appear when both sides of an equation are squared, as it may introduce additional roots that weren't solutions to the initial equation. A common example is when x is squared, it can't differentiate between positive and negative square roots. Here’s an example: if you solve the equation \((x-3)^2 = 9\), you may end up with \(x = 6\) or \(x = 0\). Always verify these outcomes in the initial equation.

The Importance of Checking Solutions

Checking your solutions when solving square root equations is key to ensuring accuracy. Finding a solution might not always mean you've found the right one. Because the process sometimes introduces extraneous solutions, verifying them against the original equation is crucial. To check, simply substitute your solution back into the original equation to see if it holds true. If it doesn't, it means the solution isn't valid. In our example \(14 = \sqrt{x}\), after finding \(x = 196\), you'd plug 196 back in: \(14 = \sqrt{196}\). Simplifying this gives you \(14 = 14\), confirming 196 is indeed the correct solution. This step ensures you haven't picked up any extra, incorrect solutions during your calculations.

The Process of Equation Isolation

Equation isolation is critical when solving equations with square roots because it simplifies the problem. To successfully isolate a square root, ensure it’s alone on one side of the equation. This sets up the problem for an easy removal of the root by squaring.In our example, the square root was already isolated: \(14 = \sqrt{x}\). If it wasn't, you would need to rearrange the equation to achieve this before proceeding. Once isolated, you can square both sides to eliminate the square root entirely and solve for the unknown variable.Isolation is essential because it allows for a straightforward step to get rid of the root and work with simpler arithmetic. It's always a preference in solving that makes the subsequent steps much easier and manageable.