Problem 9 - Elementary and Intermediate Algebra

Question

Solve. $$\sqrt{3 x}+1=6$$

Step-by-Step Solution

Answer

x = \frac{25}{3}

Step 1: Isolate the square root

Firstly, subtract 1 from both sides of the equation to isolate the square root term. \( \sqrt{3x} + 1 - 1 = 6 - 1 \)This simplifies to: \( \sqrt{3x} = 5 \)

Step 2: Square both sides

Next, square both sides to eliminate the square root. \ \( (\sqrt{3x})^2 = 5^2 \)This results in: \( 3x = 25 \)

Step 3: Solve for x

Now, divide both sides by 3 to solve for x. \( \frac{3x}{3} = \frac{25}{3} \)So, the value of x is: \( x = \frac{25}{3} \)

Key Concepts

Isolating the Square Root

To solve an equation involving a square root, the first step is to isolate the square root term on one side of the equation. In our example, the original equation is:
\[ \sqrt{3x} + 1 = 6 \]
To isolate \(\sqrt{3x}\), we need to get rid of the constant term on the same side. We do this by subtracting 1 from both sides of the equation:
\[ \sqrt{3x} + 1 - 1 = 6 - 1 \]
This subtraction leaves us with:
\[ \sqrt{3x} = 5 \]
Now, the square root is isolated, and we're ready to move on to the next step.

Squaring Both Sides

After isolating the square root, the next step is to eliminate it. We do this by squaring both sides of the equation. Squaring the square root of a number returns the number itself. Let's square both sides of our isolated equation:
\[ (\backslash\backslash sqrt {3x}) ^ 2 = 5 ^ 2 \]
Applying the square root rule \((\sqrt{a})^2 = a\), we get:
\[ 3x = 25 \]
Now, the equation no longer has a square root, and we're left with a linear equation. The final step is to solve for the variable \(x\).

Solving for x

With the square root term removed, we have a simple linear equation. Our goal is to solve for \(x\). The equation we have now is:
\[ 3x = 25 \]
To isolate \(x\), divide both sides by 3:
\[ \frac{3x}{3} = \frac{25}{3} \]
This simplifies to:
\[ x = \frac{25}{3} \]
Thus, the solution to the equation \(\sqrt{3x} + 1 = 6\) is:
\[ x = \frac{25}{3} \]
And there you have it! By isolating the square root, squaring both sides, and solving for \(x\), we've successfully solved the equation.