Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation algebraically. Check your solutions. $$3 \sqrt{x}-6=0$$

Step-by-Step Solution

Answer

The solution for the given equation \(3 \sqrt{x}-6=0\) is \(x=4\).

Step 1: Write down the given equation

The given equation is \(3 \sqrt{x}-6=0\)

Step 2: Isolate the square root term

\(\sqrt{x}\) can be isolated by moving the term '-6' to the other side of the equation which gives \(\sqrt{x}=2\)

Step 3: Remove the square root

Squaring both sides of the equation, we get \(x = 2^{2}=4\)

Step 4: Validate the solution

Inserting x = 4 back into the original equation, the equation does hold true, thus x = 4 is a valid solution.

Key Concepts

Isolating the Square Root

When solving square root equations, the first key step is to isolate the square root expression. This might seem complex, but it's similar to just getting the 'x' by itself in simpler algebra. To isolate the square root, we perform algebraic operations that reverse what's been done to the square root in the equation.

For our equation, we start with \(3 \sqrt{x}-6=0\). Here, the square root term, \(\sqrt{x}\), is multiplied by 3 and then 6 is subtracted. To isolate \(\sqrt{x}\), we need to 'undo' the subtraction first. We add 6 to both sides of the equation, which gives us \(3 \sqrt{x} = 6\). Next, we divide both sides by 3 to finally get \(\sqrt{x}=2\).

It's essential to keep the equation balanced, which means whatever operation we do to one side, we must do to the other as well. Isolating the square root sets up the equation for the next steps of solving.

Checking Algebraic Solutions

After solving an equation, verifying your solution is a crucial step. Sometimes, when we manipulate equations, especially with square roots, we can introduce what are called 'extraneous solutions'. These are results that mathematically fit the manipulated equation but don't satisfy the original equation.

To check our solution, we substitute the value back into the original equation and simplify. If we end up with a true statement, then our solution is correct. In our example, we found that \(x=4\). Plugging \(x\) back into \(3 \sqrt{x}-6=0\), we get \(3 \sqrt{4} - 6\), which simplifies to \(3*2 - 6 = 0\). That simplifies down to \(6 - 6 = 0\), verifying that our solution, \(x=4\), is indeed correct.

Square Root Equation Steps

To tame the complexities of square root equations, it's best to have a step-by-step approach. Here are the general steps, applied to the example of \(3 \sqrt{x}-6=0\).

Step 1: Set Up Your Equation

Write the original equation and make sure all terms are correctly aligned. With our exercise, this means simply writing down what's given.

Step 2: Isolate the Square Root

Using algebra, we rearrange the equation to have the square root term by itself on one side. Adding 6 and then dividing by 3 achieves this goal in our case.

Step 3: Remove the Square Root

We square both sides to eliminate the square root, making sure to square the entire side of the equation, not just the square root term itself. For our exercise, squaring 2 gives us \(x=4\).

Step 4: Validate the Solution

The last step is crucial - plug your solution back into the original equation to ensure it works. Any solution that doesn't check out isn't a true solution to the original problem.