The root of the equation `17x^(2)+17x tan [2 tan^(-1)((1)/(5))-(pi)/(4)]-10=0` is

Options

A. `(10)/(7)`
B. `-1`
C. `-(7)/(17)`
D. 1

Answer

To solve the equation \( 17x^2 + 17x \tan\left[2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right] - 10 = 0 \), we will follow these steps:

### Step 1: Simplify the expression inside the tangent function
We start with the expression \( 2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4} \).

Using the double angle formula for tangent:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Let \( \theta = \tan^{-1}\left(\frac{1}{5}\right) \), then \( \tan(\theta) = \frac{1}{5} \).

Calculating \( \tan(2\theta) \):
\[ \tan(2\tan^{-1}\left(\frac{1}{5}\right)) = \frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} = \frac{\frac{2}{5}}{1 - \frac{1}{25}} = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{2 \cdot 25}{5 \cdot 24} = \frac{10}{24} = \frac{5}{12} \]

Now we can rewrite the expression:
\[ \tan\left[2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right] = \frac{\tan(2\tan^{-1}\left(\frac{1}{5}\right)) - \tan\left(\frac{\pi}{4}\right)}{1 + \tan(2\tan^{-1}\left(\frac{1}{5}\right)) \tan\left(\frac{\pi}{4}\right)} \]
Substituting \( \tan\left(\frac{\pi}{4}\right) = 1 \):
\[ = \frac{\frac{5}{12} - 1}{1 + \frac{5}{12} \cdot 1} = \frac{\frac{5}{12} - \frac{12}{12}}{1 + \frac{5}{12}} = \frac{-\frac{7}{12}}{\frac{17}{12}} = -\frac{7}{17} \]

### Step 2: Substitute back into the original equation
Now, substituting this back into the original equation:
\[ 17x^2 + 17x\left(-\frac{7}{17}\right) - 10 = 0 \]
This simplifies to:
\[ 17x^2 - 7x - 10 = 0 \]

### Step 3: Solve the quadratic equation
We will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 17 \), \( b = -7 \), and \( c = -10 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 17 \cdot (-10) = 49 + 680 = 729 \]
Now, substituting into the quadratic formula:
\[ x = \frac{7 \pm \sqrt{729}}{34} = \frac{7 \pm 27}{34} \]
Calculating the two possible values for \( x \):
1. \( x = \frac{34}{34} = 1 \)
2. \( x = \frac{-20}{34} = -\frac{10}{17} \)

### Step 4: Conclusion
The roots of the equation are \( x = 1 \) and \( x = -\frac{10}{17} \).