Solve the following equation :
`x-(18)/(x)=6`. Give your answer correct to two significant figures.

Answer

To solve the equation \( x - \frac{18}{x} = 6 \), we will follow these steps:

### Step 1: Rearranging the equation
Start by rewriting the equation:
\[ x - \frac{18}{x} = 6 \]
To eliminate the fraction, multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 - 18 = 6x \]

### Step 2: Forming a quadratic equation
Rearranging the equation gives:
\[ x^2 - 6x - 18 = 0 \]
This is now a standard quadratic equation in the form \( ax^2 + bx + c = 0 \) where \( a = 1 \), \( b = -6 \), and \( c = -18 \).

### Step 3: Using the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-18)}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{6 \pm \sqrt{36 + 72}}{2} \]
\[ x = \frac{6 \pm \sqrt{108}}{2} \]

### Step 4: Simplifying the square root
We can simplify \( \sqrt{108} \):
\[ \sqrt{108} = \sqrt{36 \cdot 3} = 6\sqrt{3} \]
Now substituting back:
\[ x = \frac{6 \pm 6\sqrt{3}}{2} \]
This simplifies to:
\[ x = 3 \pm 3\sqrt{3} \]

### Step 5: Calculating the roots
Now we will calculate the two possible values for \( x \):
1. \( x = 3 + 3\sqrt{3} \)
2. \( x = 3 - 3\sqrt{3} \)

Using \( \sqrt{3} \approx 1.732 \):
1. For \( x = 3 + 3 \times 1.732 \):
\[ x \approx 3 + 5.196 \approx 8.196 \]
2. For \( x = 3 - 3 \times 1.732 \):
\[ x \approx 3 - 5.196 \approx -2.196 \]

### Step 6: Rounding to two significant figures
Now rounding both values to two significant figures:
1. \( 8.196 \) rounds to \( 8.2 \)
2. \( -2.196 \) rounds to \( -2.2 \)

### Final Answer
Thus, the solutions to the equation \( x - \frac{18}{x} = 6 \) are:
\[ x \approx 8.2 \quad \text{and} \quad x \approx -2.2 \]