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Chapter 8: Problem 11

(a) solve. (b) check. $$\sqrt{x}-1=4$$

Short Answer

Expert verified
The solution is x = 25.

Step by step solution

01

- Isolate the Square Root

Add 1 to both sides of the equation to isolate the square root term. This results in: \[ \sqrt{x} - 1 + 1 = 4 + 1 \]Thus, we have:\[ \sqrt{x} = 5 \]
02

- Eliminate the Square Root

Square both sides of the equation to eliminate the square root. This gives:\[ (\sqrt{x})^2 = 5^2 \]So, \[ x = 25 \]
03

- Check the Solution

Substitute the value of x back into the original equation to verify the solution. We check:\[ \sqrt{25} - 1 = 4 \]Simplifying, we get:\[ 5 - 1 = 4 \]which is true. Therefore, the solution is verified.

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Key Concepts

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

isolating square roots
Understanding how to isolate square roots is an important first step in solving square root equations. The general idea is to get the square root by itself on one side of the equation so you can then deal with it more easily.
Let's start simple. Suppose you have an equation with a square root term, for example, \( \sqrt{x} - 1 = 4 \)\. Your goal is to make sure that the square root term is by itself.
In our example, you'd add 1 to both sides of the equation: \[ \sqrt{x} - 1 + 1 = 4 + 1 \]
Now the equation looks like this: \[ \sqrt{x} = 5 \]
By isolating the square root, you make it easier to solve the equation in the next step.
squaring both sides
Once you've isolated the square root, the next step is to eliminate it. The way to do this is by squaring both sides of the equation.
In mathematics, squaring something means to multiply it by itself. When you square a square root, it
verifying solutions in algebra
After solving the equation by isolating and squaring, it's crucial to verify that your solution is correct. This is done by plugging the solution back into the original equation.
Let's use our solution from previous steps. We found that \( x = 25 \). We substitute this value back into the original equation: \[ \sqrt{25} - 1 = 4 \]
Simplifying, the equation becomes: \[ 5 - 1 = 4 \]
Since the left side matches the right side, the solution is correct.
  • Always plug your solution back into the original equation to confirm its validity.
  • Verifying not only ensures accuracy but also boosts your confidence in solving algebraic equations.
This final step of verification is essential for confirming that you haven't made any mistakes during solving.

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Most popular questions from this chapter

Simplify. $$\frac{a^{\frac{5}{8}}}{a^{\frac{1}{8}}}$$

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Rewrite the radical expression in exponential notation. $$\sqrt[3]{x^{2} y}$$

The length of a rectangle is \(3 \mathrm{ft}\) more than twice its width. The perimeter of the rectangle is \(66 \mathrm{ft}\). Use a system of linear equations to find its length and width.

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