If the roots of a quadratic equation are m + n and m - n, then the quadratic equation will be :

Options

A. A. `x^(2) + 2mx + m^(2) - mn + n^(2) = 0`
B. B. `x^(2) + 2mx + (m-n)^(2) = 0`
C. C. `x^(2) - 2mx + m^(2) - n^(2) = 0`
D. D. `x^(2) + 2mx + m^(2) - n^(2) = 0`

Answer

To find the quadratic equation whose roots are \( m + n \) and \( m - n \), we can use the fact that a quadratic equation can be expressed in the form:

\[ x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \]

### Step 1: Identify the roots
The roots given are:
- \( \alpha = m + n \)
- \( \beta = m - n \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) is calculated as follows:

\[ \alpha + \beta = (m + n) + (m - n) = m + n + m - n = 2m \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) is calculated as follows:

\[ \alpha \beta = (m + n)(m - n) = m^2 - n^2 \]

### Step 4: Write the quadratic equation
Now, substituting the sum and product of the roots into the quadratic equation form:

\[ x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \]

This gives us:

\[ x^2 - (2m)x + (m^2 - n^2) = 0 \]

### Final Quadratic Equation
Thus, the quadratic equation is:

\[ x^2 - 2mx + (m^2 - n^2) = 0 \]

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