Solve each of the following equatins :
`5x-(35)/(x)=18,xne0`

Answer

To solve the equation \( 5x - \frac{35}{x} = 18 \) where \( x \neq 0 \), we can follow these steps:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by \( x \) (since \( x \neq 0 \)):
\[ 5x^2 - 35 = 18x \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ 5x^2 - 18x - 35 = 0 \]

### Step 3: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = 5 \)
- \( b = -18 \)
- \( c = -35 \)

### Step 4: Apply 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 \( c \):
\[ x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4 \cdot 5 \cdot (-35)}}{2 \cdot 5} \]

### Step 5: Simplify the expression
Calculating \( b^2 - 4ac \):
\[ (-18)^2 = 324 \]
\[ 4 \cdot 5 \cdot (-35) = -700 \quad \text{(note that this is a negative term)} \]
Thus,
\[ b^2 - 4ac = 324 + 700 = 1024 \]

### Step 6: Substitute back into the formula
Now substituting back into the quadratic formula:
\[ x = \frac{18 \pm \sqrt{1024}}{10} \]

### Step 7: Calculate the square root
Since \( \sqrt{1024} = 32 \):
\[ x = \frac{18 \pm 32}{10} \]

### Step 8: Solve for the two possible values of \( x \)
Calculating the two cases:
1. \( x = \frac{18 + 32}{10} = \frac{50}{10} = 5 \)
2. \( x = \frac{18 - 32}{10} = \frac{-14}{10} = -\frac{7}{5} \)

### Final Solutions
Thus, the solutions for the equation are:
\[ x = 5 \quad \text{and} \quad x = -\frac{7}{5} \]

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