91Ó°ÊÓ

When solving square root equations such as \(\sqrt{6x} = 3\), removing the square root is the first step. This is done by squaring both sides of the equation, an operation that must be applied equally to prevent altering the equation's balance. The process is as follows: taking the square root function \(\sqrt{6x}\) and squaring it results in \(6x\), because the square root and square are inverse operations. Likewise, squaring the other side of the equation transforms \(3\) into \(3^2\) or \(9\). After squaring both sides, the equation simplifies to \(6x = 9\).

It is essential to carefully perform this operation because squaring both sides affects every term inside the square root—implicitly applying the distributive property if there's more than one term. In our example, we had only one term, which made the process straightforward.

Having removed the square root by squaring the equation, the next step is to isolate the variable \(x\). Isolation refers to getting the variable by itself on one side of the equation to solve for its value. In the context of our example, where we have \(6x = 9\), isolation of \(x\) requires us to undo the multiplication by \(6\). We achieve this by dividing both sides of the equation by \(6\).

This results in \(x = \frac{9}{6}\), which simplifies to \(x = \frac{3}{2}\) or \(1.5\).

It's crucial to perform the same operation (division, in this case) on both sides to maintain the equation's equality. Sometimes, additional steps are necessary if the variable appears more than once or if there are more complex terms. The goal is to rewrite the equation such that \(x\) appears alone on one side.

Even though algebraic manipulations should give us the right answer, it's imperative to check our solutions. For square root equations, particularly, checking helps to avoid extraneous solutions—answers that result from the algebraic process but do not satisfy the original equation. To check the solution \(x = \frac{3}{2}\) in our problem \(\sqrt{6x} = 3\), we replace \(x\) with \(\frac{3}{2}\) and see if the equation holds true.

The check yields \( \sqrt{6*\frac{3}{2}} = 3 \) which simplifies to \( \sqrt{9} = 3 \). Since the square root of \(9\) is indeed \(3\), both sides of our original equation agree, confirming the solution is correct. It's a good habit to routinely perform this check with every algebra problem to ensure accuracy and deepen understanding of the concepts involved.