The variance of the numbers 2, 3, 11 and x is `(49)/(4),` then the value of x are

Options

A. 6 or `(14)/(3)`
B. `6 or (14)/(5)`
C. `6 or (16)/(3)`
D. `4 or (13)/(5)`

Answer

To find the value of \( x \) such that the variance of the numbers 2, 3, 11, and \( x \) is \( \frac{49}{4} \), we will follow these steps:

### Step 1: Understand the formula for variance
The variance \( \sigma^2 \) of a set of numbers is given by the formula:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
where \( n \) is the number of elements in the set.

### Step 2: Identify the values
In our case, we have the numbers 2, 3, 11, and \( x \). Thus, we have:
- \( n = 4 \)
- The numbers are \( x_1 = 2, x_2 = 3, x_3 = 11, x_4 = x \)

### Step 3: Calculate \( \sum x_i \) and \( \sum x_i^2 \)
1. Calculate \( \sum x_i \):
\[ \sum x_i = 2 + 3 + 11 + x = 16 + x \]

2. Calculate \( \sum x_i^2 \):
\[ \sum x_i^2 = 2^2 + 3^2 + 11^2 + x^2 = 4 + 9 + 121 + x^2 = 134 + x^2 \]

### Step 4: Substitute into the variance formula
We know that the variance is given as \( \frac{49}{4} \). Therefore, we can set up the equation:
\[ \frac{134 + x^2}{4} - \left(\frac{16 + x}{4}\right)^2 = \frac{49}{4} \]

### Step 5: Simplify the equation
1. Calculate \( \left(\frac{16 + x}{4}\right)^2 \):
\[ \left(\frac{16 + x}{4}\right)^2 = \frac{(16 + x)^2}{16} = \frac{256 + 32x + x^2}{16} \]

2. Substitute back into the variance equation:
\[ \frac{134 + x^2}{4} - \frac{256 + 32x + x^2}{16} = \frac{49}{4} \]

### Step 6: Clear the fractions by multiplying through by 16
\[ 16 \left(\frac{134 + x^2}{4}\right) - (256 + 32x + x^2) = 49 \cdot 4 \]
This simplifies to:
\[ 4(134 + x^2) - (256 + 32x + x^2) = 196 \]
\[ 536 + 4x^2 - 256 - 32x - x^2 = 196 \]
\[ 3x^2 - 32x + 80 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 3, b = -32, c = 80 \).

Calculate the discriminant:
\[ b^2 - 4ac = (-32)^2 - 4 \cdot 3 \cdot 80 = 1024 - 960 = 64 \]
Now, substitute into the quadratic formula:
\[ x = \frac{32 \pm \sqrt{64}}{6} = \frac{32 \pm 8}{6} \]
This gives us two solutions:
1. \( x = \frac{40}{6} = \frac{20}{3} \)
2. \( x = \frac{24}{6} = 4 \)

### Final Values of \( x \)
Thus, the values of \( x \) are \( \frac{20}{3} \) and \( 4 \).

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