91Ó°ÊÓ

Square root equations involve expressions with square roots, often challenging for students at first. These equations typically have the following form: one or more terms contain square root symbols, like \(\sqrt{x}\).
In our problem, the equation \(\sqrt{x} - \sqrt{3x-3} = 1\) presents two square root terms. These need careful handling to solve the equation correctly.
When solving square root equations, the goal is to isolate the square root term and then eliminate the square root by squaring both sides of the equation. This approach transforms the equation into a simpler algebraic form that can be solved using standard algebraic techniques.

The isolation technique is a fundamental algebraic method in solving equations, especially effective for square root equations. It involves rearranging the terms in the equation to "isolate" a particular term on one side of the equation.
For our given equation, the initial goal was to isolate \(\sqrt{3x-3}\) by manipulating the equation. This was achieved by moving the term to the other side, resulting in \(\sqrt{x} = 1 + \sqrt{3x-3}\).
By isolating the square root, it provides a clearer path to solve further by removing the square root itself through subsequent steps. The key is to carefully perform each manipulation step to maintain equation balance.

To eliminate the square roots in an equation, the method of squaring both sides is crucial. By squaring both sides, you effectively remove the square root sign on the isolated term.
For example, in the equation \(\sqrt{x} = 1 + \sqrt{3x-3}\), squaring both sides yields \(x = (1 + \sqrt{3x-3})^2\). This step transforms the equation into a more easily solvable algebraic expression.
It's important to apply the squaring operation meticulously. Carefully expand expressions after squaring to avoid errors. Be mindful that squaring both sides can introduce extraneous solutions, which means it's important to verify potential solutions in the original equation.

Sometimes, solving equations involving square roots can lead to a conclusion that there is no real solution. This happens if, after solving the equation, the solution does not satisfy the original equation when substituted back.
In this case, solving resulted in \(x = -1\). Substituting \(-1\) back into the original equation gave an undefined term \(\sqrt{-1}\), which has no real number solution since square roots of negative numbers are not defined in real numbers.
This example highlights the importance of checking your solution by plugging it back into the original equation. Ensuring the solution is valid prevents incorrect conclusions, as observed in this problem where there is no real solution.