91Ó°ÊÓ

Solving equations is like solving a puzzle where you try to find the value of the unknown, usually denoted as \( x \). In the given exercise, the unknown is buried inside a square root, making it a bit trickier. To solve the equation \( \sqrt{1-3x} = 2 \), the first move is to eliminate the square root.
This typically involves squaring both sides of the equation. By doing this, the square root on the left is "undone," simplifying the equation to a more manageable form: \( 1 - 3x = 4 \). With the square root gone, you are left with a linear equation.
\
Next, isolate \( x \) by performing arithmetic operations. Subtract \( 1 \) from both sides, then divide by \(-3\) to solve for \( x \). Always remember the goal: you want to get \( x \) by itself on one side of the equation so you know exactly what value it takes. This systematic method is key to solving most algebraic equations.

Extraneous solutions can pop up when dealing with equations involving square roots or other functions that aren't one-to-one. But, what exactly are extraneous solutions? They are solutions that seem correct when you solve mathematically, but don't actually satisfy the original equation.
In our exercise, after squaring both sides to eliminate the square root, it might appear like any \( x \) found would be a valid solution. However, it's vital to check each potential answer by substituting it back into the original equation. This ensures the left-hand side equals the right-hand side, confirming its validity.
In this case, plugging \( x = -1 \) back into the original equation results in \( \sqrt{4} = 2 \), which is true. Thus, \( x = -1 \) is not extraneous, but a valid solution. Always conduct this critical last step to validate your answers.

Square root equations involve finding unknowns that are locked inside a square root. The main challenge here is to get rid of the square root so that you can solve for the variable straight forward.
The common method, as seen in our example, is to square both sides of the equation. This allows you to remove the square root and simplify the problem. However, it's crucial to remember that squaring both sides can introduce extraneous solutions. Why does this happen? Because squaring eliminates any sign, it might falsely imply some numbers fit the equation when they don't actually satisfy the original condition.
Always go back to check if the solution still holds true when substituted into the original equation. This double-check ensures that you haven't caught an extraneous solution in your process, preserving the integrity of your mathematical solution.