Problem 51 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \sqrt{6 t+9}=3 \sqrt{t} $$

Step-by-Step Solution

Answer

The solution is \( t = 3 \).

Step 1: Isolate the Radicals

Given the equation is \( \sqrt{6t + 9} = 3\sqrt{t} \). Both sides of the equation contain radicals. Since they are isolated, we can proceed to the next step.

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square roots. Squaring \( \sqrt{6t + 9} \) results in \( 6t + 9 \), and squaring \( 3\sqrt{t} \) results in \( 9t \).So the equation becomes: \[ 6t + 9 = 9t \]

Step 3: Simplify the Equation

Rearrange the equation by moving all terms involving \( t \) to one side:\[ 6t + 9 = 9t \]Subtract \( 6t \) from both sides to isolate \( t \):\[ 9 = 9t - 6t \]This simplifies to:\[ 9 = 3t \]

Step 4: Solve for \( t \)

To find \( t \), divide both sides of the equation by 3:\[ \frac{9}{3} = \frac{3t}{3} \]This simplifies to:\[ t = 3 \]

Step 5: Verify the Solution

Substitute \( t = 3 \) back into the original equation to ensure it satisfies it:\[ \sqrt{6(3) + 9} = 3\sqrt{3} \]Calculating the left side:\[ \sqrt{18 + 9} = \sqrt{27} = 3\sqrt{3} \]Both sides equal \( 3\sqrt{3} \), confirming that \( t = 3 \) is the correct solution.

Key Concepts

Square Roots

Square roots are mathematical expressions often represented by the symbol \( \sqrt{} \). They are used to determine a number which, when multiplied by itself, will yield the original number under the square root symbol. For instance, the square root of 16 is 4, because \(4 \times 4 = 16\). This is notated as \( \sqrt{16} = 4 \).

When solving equations involving square roots, the main challenge is to eliminate the square root by squaring. This allows us to convert the equation into a polynomial form, where regular algebraic techniques can be applied. For example, if you have \( \sqrt{6t + 9} = 3\sqrt{t} \), squaring both sides helps to remove the square roots, letting you work with linear expressions instead.

• Ensure the expression under the square root is non-negative.
• Use the property: \( (\sqrt{x})^2 = x \) to remove square roots effectively.
• Check if you can simplify the expression under the square root for easier computation.

Isolating Radicals

Isolating radicals involves rearranging an equation such that one term with a radical is on one side, and everything else on the opposite side. This is an essential step when both sides of the equation contain radicals. It sets the stage for effectively applying algebraic operations like squaring to eliminate the radicals.

In the exercise, the equation \( \sqrt{6t + 9} = 3\sqrt{t} \) already has the radicals isolated—one radical expression on each side. Therefore, the equation can be moved to the next step. In cases where they might not be isolated, you need to:

• Rearrange the equation to isolate one radical.
• Keep the equation balanced by performing identical operations on both sides.
• Only proceed to square both sides once the radicals are isolated.

This ensures clarity and prevents errors in further steps.

Verification of Solution

After obtaining a potential solution from solving the equation, it's crucial to verify that this solution indeed satisfies the original equation. This step helps to confirm the validity of the solution, ensuring there are no extraneous roots, which may arise from the squaring process.

For the exercise equation \( \sqrt{6t + 9} = 3\sqrt{t} \) with \( t = 3 \) as the solution, substituting back into the original equation is necessary:

• Calculate both sides of the original equation with \( t = 3 \): \( \sqrt{6(3) + 9} = 3\sqrt{3} \).
• The left side simplifies to \( \sqrt{27} = 3\sqrt{3} \), which matches the right side.
• Ensure both sides are equal, confirming \( t = 3 \) as a valid solution.

This verification step is vital because squaring both sides of an equation can sometimes introduce solutions that aren't valid in the original context.