If the roots of `x^(4)+qx^(2)+kx+225=0` are in arthmetic progression, then the value of `q`, is

Options

A. `15`
B. `25`
C. `35`
D. `-50`

Explanation

To solve the problem, we need to find the value of \( q \) in the polynomial equation \( x^4 + qx^2 + kx + 225 = 0 \) given that the roots are in arithmetic progression (AP).

### Step-by-Step Solution:

1. **Define the Roots**:
Since the roots are in AP, we can denote them as:
\[ r_1 = x - 3y, \quad r_2 = x - y, \quad r_3 = x + y, \quad r_4 = x + 3y \]

2. **Sum of the Roots**:
The sum of the roots can be calculated as:
\[ r_1 + r_2 + r_3 + r_4 = (x - 3y) + (x - y) + (x + y) + (x + 3y) = 4x \]
According to Vieta's formulas, the sum of the roots for the polynomial \( x^4 + qx^2 + kx + 225 = 0 \) is equal to 0 (since there is no \( x^3 \) term). Therefore:
\[ 4x = 0 \implies x = 0 \]

3. **Substituting \( x \)**:
Substituting \( x = 0 \) into the roots gives:
\[ r_1 = -3y, \quad r_2 = -y, \quad r_3 = y, \quad r_4 = 3y \]

4. **Product of the Roots**:
The product of the roots can be calculated as:
\[ r_1 \cdot r_2 \cdot r_3 \cdot r_4 = (-3y)(-y)(y)(3y) = 9y^4 \]
According to Vieta's formulas, the product of the roots for the polynomial \( x^4 + qx^2 + kx + 225 = 0 \) is equal to \( 225 \). Therefore:
\[ 9y^4 = 225 \]

5. **Solving for \( y^4 \)**:
Dividing both sides by 9 gives:
\[ y^4 = \frac{225}{9} = 25 \]
Taking the fourth root:
\[ y = \sqrt[4]{25} = \sqrt{5} \]

6. **Finding \( q \)**:
Now, we need to find \( q \). The sum of the products of the roots taken two at a time (which corresponds to \( q \)) is given by:
\[ r_1 r_2 + r_1 r_3 + r_1 r_4 + r_2 r_3 + r_2 r_4 + r_3 r_4 \]
Calculating these:
- \( r_1 r_2 = (-3y)(-y) = 3y^2 \)
- \( r_1 r_3 = (-3y)(y) = -3y^2 \)
- \( r_1 r_4 = (-3y)(3y) = -9y^2 \)
- \( r_2 r_3 = (-y)(y) = -y^2 \)
- \( r_2 r_4 = (-y)(3y) = -3y^2 \)
- \( r_3 r_4 = (y)(3y) = 3y^2 \)

Adding these together:
\[ 3y^2 - 3y^2 - 9y^2 - y^2 - 3y^2 + 3y^2 = -10y^2 \]

Substituting \( y^2 = 5 \):
\[ -10y^2 = -10 \cdot 5 = -50 \]
Therefore, \( q = -50 \).

### Final Answer:
\[ \boxed{-50} \]