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Chapter 1: Problem 10

Find the real solutions, if any, of each equation. $$ \sqrt{3 t+4}=2 $$

Short Answer

Expert verified
The solution is \( t = 0 \).

Step by step solution

01

- Isolate the Square Root

The given equation is \( \sqrt{3t + 4} = 2 \). Since the square root is already isolated on one side of the equation, we can proceed to the next step.
02

- Square Both Sides

To eliminate the square root, square both sides of the equation: \( (\sqrt{3t + 4})^2 = 2^2 \). This simplifies to: \( 3t + 4 = 4 \).
03

- Solve for t

To solve for \( t \), subtract 4 from both sides of the equation: \( 3t + 4 - 4 = 4 - 4 \). This simplifies to: \( 3t = 0 \). Now, divide both sides by 3: \( t = \frac{0}{3} \). This simplifies to: \( t = 0 \).
04

- Verify the Solution

Substitute \( t = 0 \) back into the original equation to verify it: \( \sqrt{3(0) + 4} = \sqrt{4} = 2 \). Since both sides of the equation are equal, \( t = 0 \) is a valid solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Verify the Solution
After squaring both sides and solving for the variable, the last step is verification. When we solved \( 3t + 4 = 4 \), this simplified to \( t=0\). We need to plug this value back into the original equation \( \(\sqrt{3(0) + 4} = 2 \) \). This simplifies to: \( \sqrt{4} = 2 \) \. Since both sides match, \( t = 0 \) is indeed a solution.
This verification step ensures that:
  • The solution satisfies the original equation.
  • No errors were made in the algebraic manipulation.
Sometimes squaring both sides can introduce a false solution, so re-checking is essential to confirm the root(s).

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