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Chapter 3: Problem 12

Solve each equation. If it has no solution, write "no solution." $$ 5 \sqrt{x}=25 $$

Short Answer

Expert verified
The solution is \( x = 25 \).

Step by step solution

01

Isolate the Square Root

First, isolate the square root on one side of the equation. In this case, it is already isolated: \( 5 \sqrt{x} = 25 \)
02

Divide by the Coefficient

Divide both sides of the equation by the coefficient of the square root, which is 5, to get: \( \sqrt{x} = \frac{25}{5} \) \ Simplify the right-hand side: \( \sqrt{x} = 5 \)
03

Square Both Sides

Eliminate the square root by squaring both sides of the equation: \( (\sqrt{x})^2 = 5^2 \) \ This simplifies to: \( x = 25 \)
04

Verify the Solution

Substitute \( x = 25 \) back into the original equation to check if it works: \( 5 \sqrt{25} = 25 \) \ Since \( \sqrt{25} = 5 \), the equation becomes: \( 5 \times 5 = 25 \) \ This is true, so \( x = 25 \) is a valid solution.

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Key Concepts

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

Isolating Square Roots
When solving equations involving square roots, the first thing you need to do is isolate the square root term. Isolating means getting the square root by itself on one side of the equation.
In the problem given, the equation is already isolated: \( 5 \sqrt{x} = 25 \). This means the square root term \( \sqrt{x} \) is by itself on one side, which makes the subsequent steps easier to handle.
Dividing Coefficients
Once the square root term is isolated, the next step is to get rid of any coefficients attached to it. Coefficients are numbers that multiply the variable or square root.
In our example, the term \( 5 \sqrt{x} \) has a coefficient of 5.
To eliminate this 5 and simplify the equation, you divide both sides by 5:
\[ \sqrt{x} = \frac{25}{5} \]
After simplifying, you get:
\[ \sqrt{x} = 5 \]
Squaring Both Sides
Next, to get rid of the square root, you square both sides of the equation. Squaring means to raise a number to the power of 2.
For our example, \(\root{x} = 5 \):
\[ (\root{x})^2 = 5^2 \]
This step is key because squaring the square root will cancel out the square root, leaving you with the variable alone.
Simplifying further, you get: \[ x = 25 \]
Verifying Solutions
The final step is to verify if the solution you obtained is correct. This is done by substituting the value back into the original equation.
For our example, substitute \x=25 \ into \( 5 \sqrt{x} = 25 \):
\[ 5 \root{25} = 25 \]
Simplifying \sqrt{25} \ gives you 5:
\[ 5 \times 5 = 25 \]
This confirms that our solution \ x = 25 \ is indeed correct. Verifying helps to make sure no mistakes were made during the process.

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