If `0^(@)lt x lt 90^(@)`, and `tan^(2)x-tan x = 6`, which of the following could be the value of x in degrees (rounded to the nearest degree) ?

Options

A. 63
B. 67
C. 72
D. 77

Explanation

To solve the equation \( \tan^2 x - \tan x = 6 \) where \( 0^\circ < x < 90^\circ \), we will follow these steps:

### Step 1: Substitute \( y = \tan x \)
We start by letting \( y = \tan x \). This allows us to rewrite the equation in terms of \( y \):
\[ y^2 - y = 6 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to set it to zero:
\[ y^2 - y - 6 = 0 \]

### Step 3: Factor the quadratic equation
We need to factor the quadratic equation \( y^2 - y - 6 = 0 \). We look for two numbers that multiply to \(-6\) and add to \(-1\). The numbers \(-3\) and \(2\) fit this requirement:
\[ (y - 3)(y + 2) = 0 \]

### Step 4: Solve for \( y \)
Now we set each factor to zero:
1. \( y - 3 = 0 \) → \( y = 3 \)
2. \( y + 2 = 0 \) → \( y = -2 \)

### Step 5: Determine valid solutions for \( x \)
Since \( y = \tan x \), we have:
1. \( \tan x = 3 \)
2. \( \tan x = -2 \)

However, since \( 0^\circ < x < 90^\circ \), we can only consider the positive solution:
\[ \tan x = 3 \]

### Step 6: Find \( x \)
To find \( x \), we take the arctangent:
\[ x = \tan^{-1}(3) \]

### Step 7: Calculate \( x \) using a calculator
Using a calculator, we find:
\[ x \approx 71.57^\circ \]

### Step 8: Round to the nearest degree
Since we need to round to the nearest degree:
\[ x \approx 72^\circ \]

### Conclusion
Thus, the value of \( x \) in degrees, rounded to the nearest degree, is:
\[ \boxed{72} \]

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