Calculate the possible values of x, if the standard deviation of the numbers 2, 3, 2x and 11 is 3.5.

Answer

To solve the problem, we need to find the possible values of \( x \) such that the standard deviation of the numbers \( 2, 3, 2x, \) and \( 11 \) is \( 3.5 \).

### Step 1: Calculate the Mean
The mean \( \bar{x} \) of the numbers can be calculated as follows:

\[ \bar{x} = \frac{2 + 3 + 2x + 11}{4} = \frac{16 + 2x}{4} = 4 + \frac{x}{2} \]

### Step 2: Set Up the Standard Deviation Formula
The formula for standard deviation \( \sigma \) is given by:

\[ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]

For our case, \( n = 4 \) and the numbers are \( 2, 3, 2x, 11 \). Thus, we can write:

\[ 3.5 = \sqrt{\frac{1}{4} \left( (2 - \bar{x})^2 + (3 - \bar{x})^2 + (2x - \bar{x})^2 + (11 - \bar{x})^2 \right)} \]

### Step 3: Square Both Sides
Squaring both sides gives:

\[ (3.5)^2 = \frac{1}{4} \left( (2 - \bar{x})^2 + (3 - \bar{x})^2 + (2x - \bar{x})^2 + (11 - \bar{x})^2 \right) \]

Calculating \( (3.5)^2 \):

\[ 12.25 = \frac{1}{4} \left( (2 - \bar{x})^2 + (3 - \bar{x})^2 + (2x - \bar{x})^2 + (11 - \bar{x})^2 \right) \]

Multiplying both sides by \( 4 \):

\[ 49 = (2 - \bar{x})^2 + (3 - \bar{x})^2 + (2x - \bar{x})^2 + (11 - \bar{x})^2 \]

### Step 4: Substitute the Mean
Substituting \( \bar{x} = 4 + \frac{x}{2} \):

1. \( (2 - (4 + \frac{x}{2}))^2 = (2 - 4 - \frac{x}{2})^2 = (-2 - \frac{x}{2})^2 = (2 + \frac{x}{2})^2 \)
2. \( (3 - (4 + \frac{x}{2}))^2 = (3 - 4 - \frac{x}{2})^2 = (-1 - \frac{x}{2})^2 = (1 + \frac{x}{2})^2 \)
3. \( (2x - (4 + \frac{x}{2}))^2 = (2x - 4 - \frac{x}{2})^2 = (2x - 4 - 0.5x)^2 = (1.5x - 4)^2 \)
4. \( (11 - (4 + \frac{x}{2}))^2 = (11 - 4 - \frac{x}{2})^2 = (7 - \frac{x}{2})^2 \)

### Step 5: Expand Each Term
Now we expand each squared term:

1. \( (2 + \frac{x}{2})^2 = 4 + 2x + \frac{x^2}{4} \)
2. \( (1 + \frac{x}{2})^2 = 1 + x + \frac{x^2}{4} \)
3. \( (1.5x - 4)^2 = 2.25x^2 - 12x + 16 \)
4. \( (7 - \frac{x}{2})^2 = 49 - 7x + \frac{x^2}{4} \)

### Step 6: Combine and Simplify
Now combine all these:

\[ 49 = \left(4 + 1 + 16 + 49\right) + \left(2x + x - 12x - 7x\right) + \left(\frac{x^2}{4} + \frac{x^2}{4} + 2.25x^2 + \frac{x^2}{4}\right) \]

This simplifies to:

\[ 49 = 70 - 16x + \frac{5.25x^2}{4} \]

### Step 7: Rearranging the Equation
Rearranging gives:

\[ 0 = 5.25x^2 - 16x + 21 \]

### Step 8: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Where \( a = 5.25, b = -16, c = 21 \):

\[ D = (-16)^2 - 4 \cdot 5.25 \cdot 21 \]

Calculating \( D \):

\[ D = 256 - 441 = -185 \]

Since the discriminant is negative, there are no real solutions for \( x \).

### Conclusion
Thus, the possible values of \( x \) such that the standard deviation of the numbers \( 2, 3, 2x, 11 \) is \( 3.5 \) do not exist.