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Chapter 10: Problem 10

Solve. $$\sqrt{2 x}-1=2$$

Short Answer

Expert verified
x = 4.5

Step by step solution

01

Isolate the Square Root

First, add 1 to both sides of the equation to isolate the square root term. This gives you: \[ \sqrt{2x} - 1 + 1 = 2 + 1 \] which simplifies to \[ \sqrt{2x} = 3 \]
02

Remove the Square Root

Next, square both sides of the equation to eliminate the square root. This results in: \[ (\sqrt{2x})^2 = 3^2 \] which simplifies to \[ 2x = 9 \]
03

Solve for x

Finally, divide both sides by 2 to solve for \( x \). This gives you: \[ x = \frac{9}{2} \] which simplifies to \[ x = 4.5 \]

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Key Concepts

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

Isolate the Square Root
In solving square root equations, the first crucial step is isolating the square root. This means making sure the square root expression is by itself on one side of the equation.
Take the equation \(\root(2x) - 1 = 2\). The goal here is to move everything not under the square root to the other side.
You'll start by adding 1 to both sides of the equation. This is because subtraction of 1 is the term currently stopping the square root from being isolated.
So, the equation becomes: \(\sqrt{2x} - 1 + 1 = 2 + 1\)
This simplifies to: \(\sqrt{2x} = 3\).
Now, the square root is isolated!
Remove the Square Root
Once the square root is isolated, the next step is to remove it. To do that, you need to square both sides of the equation. Squaring reverses the square root function. Let's consider the isolated equation from previous step: \(\sqrt{2x} = 3\)
Squaring both sides gives us: \((\sqrt{2x})^2 = 3^2\)
We know that squaring a square root cancels the square root, thus simplifying it to: \(2x = 9\).
You are now left with a simple linear equation.
Simplify the Equation
The final step is to simplify the equation to solve for \(x\). With \(2x = 9\), we need to isolate \(x\) by dividing both sides of the equation by 2.
This results in: \(2x / 2 = 9 / 2\)
Simplifying the fractions on the right gives us: \(x = 4.5\).
Now you have your solution: \(x = 4.5\).

To summarize, when solving square root equations, remember these three core steps:
  • Isolate the square root term.
  • Remove the square root by squaring both sides.
  • Simplify the resulting equation to find your answer.
Following these steps will help you tackle square root equations with ease!

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