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Chapter 1: Problem 9

Find the real solutions, if any, of each equation. $$ \sqrt{2 t-1}=1 $$

Short Answer

Expert verified
The real solution to the equation is t = 1.

Step by step solution

01

Isolate the Square Root

Ensure that the square root term is isolated on one side of the equation. In this exercise, the square root is already isolated: \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ \ \ \ \ \ \ \(\ \) \( \) as the right side value of the equation as-is.
02

Square Both Sides

Square both sides of the equation to remove the square root and obtain a solvable equation. This gives: \ \ \ \ \ \ \ \ \(\ 2t-1 = 1^2\ = \ 1 \)\. \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,\ \ \ \ \ \ \ \, \, \ \ \ \, \, is as you get the same result.
03

Solve for t

Solve for the variable t by isolating it on one side of the equation. Simplify: $$ \ 2t-1. 2t = 2$$ Hence:. $$ \ t = 1$$. \ So \ the final result is obtained. This step carefully verifies and validates to, solve the equation., Simplifying it directly..Using: t=1,,\, \,\ \,\the right answer

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Key Concepts

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

Isolating the Square Root
To solve an equation that includes a square root, we first need to isolate the square root term. This means getting the square root by itself on one side of the equation. In the given problem, \(\text{\sqrt{2t-1}=1}\), the square root is already isolated.

This step ensures that we can effectively move on to eliminating the square root in the next phase.

Always check for the square root to be alone on one side to avoid errors in subsequent steps.
Squaring Both Sides
Once the square root is isolated, we can eliminate it by squaring both sides of the equation. Squaring both sides effectively 'undoes' the square root.

For our example: \( \text{\sqrt{2t-1}=1}\), squaring both sides gives us \( \text{(\sqrt{2t-1})^2 = 1^2} = 2t-1 = 1\).

This step is crucial because it transforms the equation into a more familiar form that is easier to solve.
Solving for Variables
After squaring both sides, the resulting equation is a simpler one without any square roots. Now, we solve for the variable (in this case, \( t \)). For our equation, \( 2t-1 = 1\):

  • Add 1 to both sides to isolate the term with the variable: \( 2t - 1 + 1 = 1 + 1\).
  • This simplifies to \( 2t = 2\).
  • Divide by 2: \( t = 1\).

Thus, we find that \( t = 1\). This completes the solving process.
Verification of Solutions
After solving for the variable, it is important to verify our solution. This helps ensure that our solution is correct and that no extraneous solutions (solutions that do not satisfy the original equation) have been introduced.

Substitute \( t = 1 \) back into the original equation to check:
\( \sqrt{2(1)-1}=\sqrt{2-1}=\sqrt{1}=1 \).

The left-hand side equals the right-hand side, confirming that \( t = 1\) is indeed a valid solution. Always remember to verify to avoid errors in your final answer.

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