IF m,n,K are rational and `m=k +(n)/(k) ` then the roots of ` x^2 +mx +n =0` are

Options

A. k, `(n)/(k)`
B. k, `(-n)/(k)`
C. -k, `(-n)/(k)`
D. -k, `(n)/(k)`

Answer

To solve the problem, we need to find the roots of the quadratic equation \( x^2 + mx + n = 0 \) given that \( m = k + \frac{n}{k} \), where \( m, n, k \) are rational numbers.

### Step-by-step Solution:

1. **Substitute \( m \) into the equation**:
We start with the equation:
\[ x^2 + mx + n = 0 \]
Substituting \( m = k + \frac{n}{k} \):
\[ x^2 + \left(k + \frac{n}{k}\right)x + n = 0 \]

2. **Identify coefficients**:
Here, we identify the coefficients:
- \( a = 1 \)
- \( b = k + \frac{n}{k} \)
- \( c = n \)

3. **Use the quadratic formula**:
The roots of the quadratic equation are given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(k + \frac{n}{k}) \pm \sqrt{\left(k + \frac{n}{k}\right)^2 - 4 \cdot 1 \cdot n}}{2 \cdot 1} \]

4. **Simplify the expression**:
First, simplify the expression for \( b^2 - 4ac \):
\[ b^2 = \left(k + \frac{n}{k}\right)^2 = k^2 + 2n + \frac{n^2}{k^2} \]
Then calculate \( 4ac \):
\[ 4ac = 4n \]
Thus, we have:
\[ b^2 - 4ac = k^2 + 2n + \frac{n^2}{k^2} - 4n = k^2 + \frac{n^2}{k^2} - 2n \]

5. **Substitute back into the quadratic formula**:
Now substituting back into the formula for \( x \):
\[ x = \frac{-(k + \frac{n}{k}) \pm \sqrt{k^2 + \frac{n^2}{k^2} - 2n}}{2} \]

6. **Factor the square root**:
Notice that:
\[ k^2 + \frac{n^2}{k^2} - 2n = \left(k - \frac{n}{k}\right)^2 \]
Therefore, we can write:
\[ x = \frac{-(k + \frac{n}{k}) \pm \left(k - \frac{n}{k}\right)}{2} \]

7. **Calculate the two roots**:
- For the addition:
\[ x_1 = \frac{-k - \frac{n}{k} + k - \frac{n}{k}}{2} = \frac{-2\frac{n}{k}}{2} = -\frac{n}{k} \]
- For the subtraction:
\[ x_2 = \frac{-k - \frac{n}{k} - (k - \frac{n}{k})}{2} = \frac{-2k}{2} = -k \]

8. **Final roots**:
Thus, the roots of the quadratic equation \( x^2 + mx + n = 0 \) are:
\[ x = -\frac{n}{k} \quad \text{and} \quad x = -k \]