The gradient of the curve is given by `(dy)/(dx)=2x-(3)/(x^(2))`.
The curve passes through (1, 2) find its equation.

Answer

To find the equation of the curve given its gradient, we will follow these steps:

### Step 1: Write down the given gradient
The gradient of the curve is given by:
\[ \frac{dy}{dx} = 2x - \frac{3}{x^2} \]

### Step 2: Rearrange the equation for integration
We can rearrange the equation as:
\[ dy = \left(2x - \frac{3}{x^2}\right)dx \]

### Step 3: Integrate both sides
Now, we will integrate both sides. The left side integrates to \(y\), and we will integrate the right side term by term:
\[ \int dy = \int \left(2x - \frac{3}{x^2}\right)dx \]

### Step 4: Perform the integration
1. The integral of \(2x\) is:
\[ \int 2x \, dx = x^2 \]
2. The integral of \(-\frac{3}{x^2}\) can be rewritten as \(-3x^{-2}\):
\[ \int -3x^{-2} \, dx = -3 \cdot \left(-\frac{1}{x}\right) = \frac{3}{x} \]

Putting it all together, we have:
\[ y = x^2 + \frac{3}{x} + C \]
where \(C\) is the constant of integration.

### Step 5: Use the point (1, 2) to find C
Since the curve passes through the point (1, 2), we can substitute \(x = 1\) and \(y = 2\) into the equation:
\[ 2 = 1^2 + \frac{3}{1} + C \]
This simplifies to:
\[ 2 = 1 + 3 + C \implies 2 = 4 + C \implies C = 2 - 4 = -2 \]

### Step 6: Write the final equation of the curve
Now that we have the value of \(C\), we can write the equation of the curve:
\[ y = x^2 + \frac{3}{x} - 2 \]

### Final Answer
The equation of the curve is:
\[ y = x^2 + \frac{3}{x} - 2 \]

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