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Chapter 1: Problem 18

Find the real solutions, if any, of each equation. $$ x=3 \sqrt{x} $$

Short Answer

Expert verified
The real solutions are \( x = 0 \) and \( x = 9 \).

Step by step solution

01

Isolate the square root term

The equation given is: \[ x = 3 \sqrt{x} \]To solve for \( x \), first isolate the square root term. Rewrite the equation: \[ x = 3 \sqrt{x} \]
02

Square both sides

To eliminate the square root, square both sides of the equation: \[ x^2 = (3 \sqrt{x})^2 \] This simplifies to: \[ x^2 = 9x \]
03

Move all terms to one side

Rearrange the equation to set it to 0: \[ x^2 - 9x = 0 \]
04

Factor the quadratic equation

Factor the equation: \[ x(x - 9) = 0 \]
05

Solve for x

Set each factor equal to zero and solve: \[ x = 0 \] or \[ x = 9 \]
06

Check the solutions

Substitute the solutions back into the original equation to check for extraneous solutions. For \( x = 0 \): \[ 0 = 3 \sqrt{0} \rightarrow 0 = 0 \]For \( x = 9 \): \[ 9 = 3 \sqrt{9} \rightarrow 9 = 9 \]

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Key Concepts

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

Isolating Square Root
To solve an equation that involves a square root, the first step is usually to isolate the square root term. This makes it easier to eliminate the square root in the next steps. In the given problem, we start with the equation:
\[ x = 3 \sqrt{x} \]
To isolate the square root, let's look at the right side of the equation. It has the square root \( \sqrt{x} \) multiplied by 3. Since it's already isolated (i.e., on its own side of the equation), we are ready to proceed to the next steps where we'll get rid of the square root.
Quadratic Equation
Once we have isolated the square root, we need to eliminate it. We do this by squaring both sides of the equation. When we have the equation:
\[ x = 3 \sqrt{x}\]
Squaring both sides gives us:
\[ x^2 = (3 \sqrt{x})^2 \]
Which simplifies to:
\[ x^2 = 9x \]
Now, we move all terms to one side to get a quadratic equation form:
\[ x^2 - 9x = 0 \]
Quadratic equations are in the form \[ ax^2 + bx + c = 0 \], where a, b, and c are constants. Here, a = 1, b = -9, and c = 0.
Factoring
Factoring is a method used to solve quadratic equations. In the equation:
\[ x^2 - 9x = 0 \]
We look for common factors. Both terms ( \ x^2 \ and \ -9x \ ) share a common factor of \ x \. So, we factor out \ x \:
\[ x(x - 9) = 0 \]
Next, we set each factor equal to zero:
\[ x = 0 \]
or
\[ x - 9 = 0 \]
Solving for \ x \ gives us the solutions:
  • \ x = 0 \
  • \ x = 9 \

Finally, we plug these solutions into the original equation to verify them. Both \ x = 0 \ and \ x = 9 \ satisfy the original equation, confirming they are the correct solutions.

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