Problem 2 - Algebra A Combined Function

Question

Solve. $$ \sqrt{3 x}=3 $$

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Isolate the Square Root

The equation given is \( \sqrt{3x} = 3 \). Here, the square root is already isolated on one side, so we can proceed to the next step.

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \( (\sqrt{3x})^2 = 3^2 \).This simplifies to \( 3x = 9 \).

Step 3: Solve for x

Now solve for \( x \) by dividing both sides by 3:\( x = \frac{9}{3} \).This simplifies to \( x = 3 \).

Step 4: Verify the Solution

Substitute \( x = 3 \) back into the original equation to verify: \( \sqrt{3(3)} = 3 \).This simplifies to \( \sqrt{9} = 3 \), which is true, confirming that our solution is correct.

Key Concepts

Isolation of Square Roots

When solving square root equations, it's important to isolate the square root expression. This makes the rest of the solving process much simpler. If you look at an equation like \( \sqrt{3x} = 3 \), you'll notice that the square root is already isolated on the left side. This means there's no need to move terms around before proceeding to the next steps.

• Check if the square root is by itself on one side of the equation.
• If not, use basic algebra to move other terms to the opposite side.

Think of it as preparing the equation for squaring both sides, which is necessary to eliminate the square root. Ensuring the square root is isolated confirms that the equation is ready for transformation, making sure the algebraic operations are accurate and straightforward.

Squaring Both Sides

Once the square root is isolated, the next logical step in solving the equation is to eliminate it by squaring both sides. Why do we square the equation? Because squaring effectively removes the square root.
For instance, in our equation \( \sqrt{3x} = 3 \), you square both sides:

• The left side: \( (\sqrt{3x})^2 = 3x \)
• The right side: \( 3^2 = 9 \)

This keeps the equation balanced and leads to a simpler equation, \( 3x = 9 \). Remember, the core idea here is to cancel out the square root while preserving the equality. This step is crucial as it transforms the original equation into a form that is much easier to solve.

Verifying Solutions

After finding the solution to the equation, it's important to verify that it satisfies the original equation. This step ensures the solution is correct and accounts for any potential extraneous solutions that could arise from squaring both sides.
To verify, substitute the solution back into the original equation and check:

• Replace \( x \) with the value found (in our example, \( x = 3 \)).
• Calculate to see if it equals the original side.

For \( x = 3 \), check: \( \sqrt{3(3)} = 3 \) simplifies to \( \sqrt{9} = 3 \), which is true. Verifying is important because sometimes squaring both sides can introduce solutions that don't really fit the original equation. This final check confirms that \( x = 3 \) is the right solution.