Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. $$ \sqrt{3 t+5}=7 $$

Step-by-Step Solution

Answer

The solution is \( t = \frac{44}{3} \).

Step 1: Isolate the Radical

The equation already has the square root isolated: \( \sqrt{3t + 5} = 7 \). You don't need to perform any additional steps to get the square root by itself since it's already isolated.

Step 2: Eliminate the Square Root

Square both sides of the equation to eliminate the square root. This gives: \( (\sqrt{3t + 5})^2 = 7^2 \). Thus, the equation becomes: \( 3t + 5 = 49 \).

Step 3: Solve for t

To solve for \( t \), start by isolating \( 3t \). Subtract 5 from both sides: \( 3t + 5 - 5 = 49 - 5 \), which simplifies to \( 3t = 44 \). Then, divide both sides by 3: \( t = \frac{44}{3} \).

Step 4: Check for Extraneous Solutions

Substitute \( t = \frac{44}{3} \) back into the original equation to ensure it is a valid solution. Calculate \( 3t + 5 \): \( 3 \times \frac{44}{3} + 5 = 44 + 5 = 49 \). Verify by seeing if \( \sqrt{49} = 7 \), which is true. Thus, \( t = \frac{44}{3} \) is a valid solution.

Key Concepts

Extraneous Solutions

When solving radical equations, we often encounter what are called extraneous solutions. These are solutions that emerge from the mathematical process of solving the equation, but do not actually satisfy the original equation itself. They can be quite misleading if not carefully checked.

Extraneous solutions often arise when both sides of an equation are squared. Let's take the example given:

• The original radical equation: \( \sqrt{3t + 5} = 7 \).
• Once squared, it becomes \( 3t + 5 = 49 \).

This step can introduce solutions that do not work in the original equation. Therefore, checking the potential solutions by plugging them back into the original equation is crucial to ensure they are correct.

Always verify your solution by substituting it into the original equation before "squaring it" happened. A true solution will satisfy both the squared form and the original radical equation. This thoroughness in checking confirms whether a solution is valid, avoiding missteps due to extraneous solutions.

Square Root Isolation

To solve a radical equation effectively, isolating the square root is key. This means arranging the equation so that the square root term is on one side, while any numbers or other terms are on the opposite side.

In the equation \( \sqrt{3t + 5} = 7 \), the square root term \( \sqrt{3t + 5} \) is already isolated. It stands alone on one side of the equation, which simplifies the subsequent steps.

• When the square root is isolated, you can eliminate it by squaring both sides of the equation.
• Squaring both sides results in removing the radical, simplifying the equation to a more straightforward algebraic form.

Without this isolation step, the process becomes more complicated and solving the equation could introduce errors. Isolating the square root helps to streamline the solving process, making each subsequent algebraic manipulation straightforward and reliable.

Equation Solutions

Once the radical is eliminated, solving for the variable becomes a clearer path. Using basic algebraic principles, you can isolate and solve for the variable just like in non-radical equations.

For instance, after squaring the equation \( (\sqrt{3t + 5})^2 = 7^2 \), it turns into \( 3t + 5 = 49 \). From there:

• First, subtract 5 from both sides, simplifying the equation to \( 3t = 44 \).
• Next, divide both sides by 3 to solve for \( t \), giving \( t = \frac{44}{3} \).

Each operation maintains the balance of the equation, crucial for solving it correctly.

Upon finding solutions, as a fundamental step, check each one by substituting it back into the original radical equation. Ensuring each solution satisfies the original equation confirms its validity and dismisses any extraneous ones. This methodical cross-verification strengthens understanding and accuracy in solving radical equations.