If m + n = 14 and mn = 48 , then find the value (s) of m - n .

Answer

To solve the problem step by step, we will use the given equations and apply some algebraic identities.

### Step 1: Write down the given information
We are given:
- \( m + n = 14 \) (Equation 1)
- \( mn = 48 \) (Equation 2)

### Step 2: Square the first equation
We will square both sides of Equation 1:
\[ (m + n)^2 = 14^2 \]
This gives us:
\[ m^2 + n^2 + 2mn = 196 \]

### Step 3: Substitute the value of \( mn \)
From Equation 2, we know \( mn = 48 \). We can substitute this value into the equation we derived in Step 2:
\[ m^2 + n^2 + 2(48) = 196 \]
This simplifies to:
\[ m^2 + n^2 + 96 = 196 \]

### Step 4: Solve for \( m^2 + n^2 \)
Now, we will isolate \( m^2 + n^2 \):
\[ m^2 + n^2 = 196 - 96 \]
Calculating this gives:
\[ m^2 + n^2 = 100 \]

### Step 5: Use the identity for \( m - n \)
We want to find \( m - n \). We can use the identity:
\[ (m - n)^2 = m^2 + n^2 - 2mn \]
Substituting the values we have:
\[ (m - n)^2 = 100 - 2(48) \]
This simplifies to:
\[ (m - n)^2 = 100 - 96 \]
So we have:
\[ (m - n)^2 = 4 \]

### Step 6: Take the square root
To find \( m - n \), we take the square root of both sides:
\[ m - n = \pm 2 \]

### Final Answer
Thus, the value of \( m - n \) is \( 2 \) or \( -2 \).

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