Solve for `x` if `x^2(1/x-2)= -4`

Answer

To solve the equation \( x^2\left(\frac{1}{x} - 2\right) = -4 \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation:
\[ x^2\left(\frac{1}{x} - 2\right) = -4 \]

### Step 2: Distribute \( x^2 \)
Distributing \( x^2 \) inside the parentheses gives:
\[ x^2 \cdot \frac{1}{x} - 2x^2 = -4 \]
This simplifies to:
\[ x - 2x^2 = -4 \]

### Step 3: Rearrange the equation
Rearranging the equation to bring all terms to one side results in:
\[ -2x^2 + x + 4 = 0 \]
To make it easier to work with, we can multiply the entire equation by -1:
\[ 2x^2 - x - 4 = 0 \]

### Step 4: Identify coefficients
Now, identify the coefficients \( a \), \( b \), and \( c \) from the standard quadratic form \( ax^2 + bx + c = 0 \):
- \( a = 2 \)
- \( b = -1 \)
- \( c = -4 \)

### Step 5: Use 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 \( c \):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-4)}}{2 \cdot 2} \]

### Step 6: Simplify the expression
Calculating the discriminant:
\[ x = \frac{1 \pm \sqrt{1 + 32}}{4} \]
\[ x = \frac{1 \pm \sqrt{33}}{4} \]

### Step 7: Final solutions
Thus, the solutions for \( x \) are:
\[ x = \frac{1 + \sqrt{33}}{4} \quad \text{and} \quad x = \frac{1 - \sqrt{33}}{4} \]