The number of positive solution satisfying the equation `tan^(-1)((1)/(2x+1))+tan^(-1)((1)/(4x+1))=tan^(-1)(2/(x^2))` is

Explanation

To solve the equation

\[ \tan^{-1}\left(\frac{1}{2x+1}\right) + \tan^{-1}\left(\frac{1}{4x+1}\right) = \tan^{-1}\left(\frac{2}{x^2}\right), \]

we will use the property of the tangent inverse function:

\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \quad \text{if } ab < 1. \]

### Step 1: Apply the addition formula

Let \( a = \frac{1}{2x+1} \) and \( b = \frac{1}{4x+1} \). We can apply the addition formula:

\[ \tan^{-1}\left(\frac{1}{2x+1}\right) + \tan^{-1}\left(\frac{1}{4x+1}\right) = \tan^{-1}\left(\frac{\frac{1}{2x+1} + \frac{1}{4x+1}}{1 - \frac{1}{(2x+1)(4x+1)}}\right). \]

### Step 2: Simplify the numerator

The numerator becomes:

\[ \frac{1}{2x+1} + \frac{1}{4x+1} = \frac{(4x+1) + (2x+1)}{(2x+1)(4x+1)} = \frac{6x + 2}{(2x+1)(4x+1)}. \]

### Step 3: Simplify the denominator

The denominator becomes:

\[ 1 - \frac{1}{(2x+1)(4x+1)} = \frac{(2x+1)(4x+1) - 1}{(2x+1)(4x+1)} = \frac{8x^2 + 6x}{(2x+1)(4x+1)}. \]

### Step 4: Combine the results

Now we can combine the results:

\[ \tan^{-1}\left(\frac{6x + 2}{8x^2 + 6x}\right) = \tan^{-1}\left(\frac{2}{x^2}\right). \]

### Step 5: Set the arguments equal

Since the tangent inverse function is one-to-one, we can set the arguments equal:

\[ \frac{6x + 2}{8x^2 + 6x} = \frac{2}{x^2}. \]

### Step 6: Cross-multiply

Cross-multiplying gives:

\[ (6x + 2)x^2 = 2(8x^2 + 6x). \]

### Step 7: Expand and rearrange

Expanding both sides:

\[ 6x^3 + 2x^2 = 16x^2 + 12x. \]

Rearranging gives:

\[ 6x^3 - 14x^2 - 12x = 0. \]

### Step 8: Factor out common terms

Factoring out \( 2x \):

\[ 2x(3x^2 - 7x - 6) = 0. \]

### Step 9: Solve the quadratic equation

Now we solve the quadratic \( 3x^2 - 7x - 6 = 0 \) using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 3 \cdot (-6)}}{2 \cdot 3} = \frac{7 \pm \sqrt{49 + 72}}{6} = \frac{7 \pm \sqrt{121}}{6} = \frac{7 \pm 11}{6}. \]

### Step 10: Calculate the roots

Calculating the roots:

1. \( x = \frac{18}{6} = 3 \) (positive)
2. \( x = \frac{-4}{6} = -\frac{2}{3} \) (not positive)

### Conclusion

Thus, the number of positive solutions is **1**.