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Chapter 9: Problem 4

Solve the equation. $$\sqrt{x-2}+1=7$$

Short Answer

Expert verified
The solution to the equation is x = 38.

Step by step solution

01

Subtract 1 from both sides

To isolate the square root term on one side, we will add -1 to both sides: \( \sqrt{x-2} + 1 - 1 = 7 - 1 \), which simplifies to \( \sqrt{x-2} = 6 \).
02

Square both sides

To get rid of the square root, we will square both sides: \( (\sqrt{x-2})^2 = 6^2 \), which simplifies to \( x - 2 = 36 \).
03

Solve for x

To find x, we will add 2 to both sides: \( x - 2 + 2 = 36 + 2 \), which gives the final solution \( x = 38 \).

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Key Concepts

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

Isolating the Square Root
Isolating the square root is the initial step in solving a radical equation like \( \sqrt{x-2}+1=7 \). You need to get the square root term by itself on one side of the equation. To do this, you would look at the equation and identify any constants or terms not under the square root that you can eliminate.

Notice the \(+1\) in the equation. We need to remove it to have \( \sqrt{x-2} \) alone on one side. By subtracting 1 from both sides, we alter the equation while maintaining balance:
  • On the left side, \( \sqrt{x-2} + 1 - 1 = \sqrt{x-2} \).
  • On the right side, you have \( 7-1 = 6 \).
Now you're left with a simpler equation: \( \sqrt{x-2} = 6 \).

This process ensures that we have a clean slate to work on with our main task鈥攔emoving the square root.
Squaring Both Sides
Once you've isolated the square root, the next step involves getting rid of it entirely. The way to do this is by squaring both sides of the equation. This step is crucial since it allows us to deal with a standard algebraic equation, making it easier to solve.

By squaring both sides, you essentially eliminate the square root on the left. Here鈥檚 how it works:
  • If you have \( (\sqrt{x-2})^2 \), squaring it gives you \( x-2 \).
  • On the right side, when you square the number 6, you get \( 6^2 = 36 \).
As a result, squaring both sides of the equation \( \sqrt{x-2} = 6 \) leads to the equation \( x-2 = 36 \).

This transition from a radical form to a simpler algebraic form is critical and significantly simplifies the equation.
Algebraic Manipulation
After removing the square root, what remains is a simple algebraic equation \( x - 2 = 36 \). Now, we're in familiar territory, where basic algebraic manipulation comes into play.

The objective here is to solve for \( x \) by isolating it on one side of the equation. To achieve this, you'll need to perform the inverse operation of subtraction on \( x-2 \), which is addition.
  • Add 2 to both sides of the equation: \( x - 2 + 2 = 36 + 2 \).
  • This simplifies to \( x = 38 \).
So, the solution to the equation is \( x = 38 \), and through these basic manipulations, you've reached the answer.

This step concludes the process, showcasing how algebraic manipulation allows us to find solutions easily once the equation is free of radicals.

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Most popular questions from this chapter

Simplify the expression. $$\frac{\sqrt{x}-5}{\sqrt{x}-1}$$

Simplify the expression. $$\sqrt{x^{3} y}+4 \sqrt{x y}$$

Rationalize the denominator of the expression and simplify. $$\frac{7 \sqrt{5}}{\sqrt{3}-\sqrt{5}}$$

Vocabulary In your own words, describe a radical equation.

Rationalize the denominator of the expression and simplify. $$\frac{9}{\sqrt{x}+2}$$
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