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Chapter 3: Problem 41

Find all real numbers \(x\) that satisfy the indicated equation. $$ x-\sqrt{x}=6 $$

Short Answer

Expert verified
The solution to the equation \(x - \sqrt{x} = 6\) is \(x = 9\).

Step by step solution

01

Rearrange the given equation

First, let's rearrange the given equation to have the square root term on one side and all other terms on the other side: $$ \sqrt{x} = x - 6 $$
02

Square both sides of the equation

To eliminate the square root, let's square both sides of the equation: $$ (\sqrt{x})^2 = (x - 6)^2 $$ This simplifies to: $$ x = x^2 - 12x + 36 $$
03

Move all terms to one side and solve the quadratic equation

We want to set the equation to 0 so that we can solve the quadratic equation. To do this, let's move all terms to one side: $$ x^2 - 13x + 36 = 0 $$ Now, let's factor this quadratic equation: $$ (x - 4)(x - 9) = 0 $$ This gives us two possible solutions for x: $$ x = 4 \ \text{or} \ x = 9 $$
04

Check for extraneous roots

Finally, we need to check each solution for extraneous roots by plugging them back into the original equation: 1. \(x = 4\): $$ 4 - \sqrt{4} = 6 $$ $$ 4 - 2 \neq 6 $$ So, x = 4 is not a solution. 2. \(x = 9\): $$ 9 - \sqrt{9} = 6 $$ $$ 9 - 3 = 6 $$ So, x = 9 is a valid solution.
05

Final Answer

Therefore, the solution to the equation \(x - \sqrt{x} = 6\) is \(x = 9\).

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