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Magazine Chapter 1: Problem 43
Solve each equation. $$\sqrt{x}-\sqrt{x-5}=1$$
Short Answer
Expert verified
The solution is \(x = 9\).
Step by step solution
01
Isolate one square root
Start by isolating one of the square root terms on one side of the equation. In this case, add \(\sqrt{x-5}\) to both sides: \(\sqrt{x} = \sqrt{x-5} + 1\).
02
Square both sides
Next, square both sides of the equation to eliminate the square roots: \[ (\sqrt{x})^2 = (\sqrt{x-5} + 1)^2 \] Simplifying both sides gives: \(\ x = (x-5) + 2 \sqrt{x-5} + 1\).
03
Simplify the equation
Combine like terms on the right-hand side: \(\text{left side:} x\) = \(\text{right side:} x - 4 + 2 \sqrt{x-5}\).
04
Isolate the square root term
Subtract \(x-4\) from both sides to isolate the square root term: \(\text{left side:} 4\) = \(\text{right side:} 2 \sqrt{x-5}\).
05
Square both sides again
To get rid of the square root on the right, square both sides again: \[ (4)^2 = (2 \sqrt{x-5})^2 \] Simplifying both sides gives: \[ 16 = 4(x-5) \]
06
Solve for x
Divide both sides by 4 to solve for \(x-5\): \[ 4 = x-5 \] Add 5 to both sides to find \(x\): \(\text{left side:} 4 + 5 = 9\). Therefore, \(x = 9\).
07
Verify the solution
To verify the solution, substitute \(x = 9\) back into the original equation: \[ \sqrt{9} - \sqrt{9-5} = 1 \] Simplifying, we get: \[ 3 - 2 = 1 \] which is true. Therefore, the solution \(x = 9\) is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Isolate Square Root
To solve the radical equation ewline \( \sqrt{x} - \sqrt{x-5} = 1 \), the first step is to isolate one of the square roots. In other words, you want to get one square root term by itself on one side of the equation. For this example, we add \sqrt{x-5} to both sides, resulting in: ewline \( \sqrt{x} = \sqrt{x-5} + 1 \). ewlineBy isolating \sqrt{x}, we ease the process of further simplifying the equation.
Squaring Both Sides
Once you've isolated one of the square roots, the next step is to eliminate the square roots by squaring both sides of the equation. This action removes the radical signs. ewline In the example, we square both sides of the isolated equation: ewline \[ (\sqrt{x})^2 = (\sqrt{x-5} + 1)^2 \].ewline This gives: ewline \[ x = (x-5) + 2\sqrt{x-5} + 1 \]. After squaring, the equation includes only one square root term remaining, which simplifies the resolution of the remaining values.
Simplify Equation
Now that the equation is \( x = (x - 4) + 2 \sqrt{x-5} \), we need to combine like terms and isolate the remaining square root term. ewline Subtract \( x - 4 \) from both sides to further isolate the square root: ewline \( 4 = 2 \sqrt{x-5} \). ewline Simplify further by dividing both sides by 2 to isolate the square root: ewline \( 2 = \sqrt{x-5} \).
Verify Solution
Finally, to ensure our solution is correct, substitute \( x = 9 \) back into the original equation: ewline \[ \sqrt{9} - \sqrt{9-5} = 1 \] Simplify the radicals: ewline \[ 3 - 2 = 1 \] ewline Since the left side equals the right side of the equation, the solution \( x = 9 \) is verified and correct. Verification is crucial because squaring both sides can sometimes introduce extraneous solutions that do not satisfy the original equation.
