If `M(a, b)=a^(2)+b^(2)+ab,N(a, b)=a^(2)+b^(2)-ab`. Find the value of `M(7, N(9, 4))`.

Options

A. 5197
B. 3197
C. 4197
D. none of these

Answer

To solve the problem, we need to find the value of \( M(7, N(9, 4)) \). We will do this step by step.

### Step 1: Calculate \( N(9, 4) \)
The function \( N(a, b) \) is defined as:
\[ N(a, b) = a^2 + b^2 - ab \]
Substituting \( a = 9 \) and \( b = 4 \):
\[ N(9, 4) = 9^2 + 4^2 - 9 \cdot 4 \]
Calculating each term:
- \( 9^2 = 81 \)
- \( 4^2 = 16 \)
- \( 9 \cdot 4 = 36 \)

Now, substituting these values back into the equation:
\[ N(9, 4) = 81 + 16 - 36 \]
Calculating this step-by-step:
\[ 81 + 16 = 97 \]
\[ 97 - 36 = 61 \]
Thus, we have:
\[ N(9, 4) = 61 \]

### Step 2: Calculate \( M(7, N(9, 4)) \)
Now that we have \( N(9, 4) = 61 \), we can find \( M(7, 61) \).
The function \( M(a, b) \) is defined as:
\[ M(a, b) = a^2 + b^2 + ab \]
Substituting \( a = 7 \) and \( b = 61 \):
\[ M(7, 61) = 7^2 + 61^2 + 7 \cdot 61 \]
Calculating each term:
- \( 7^2 = 49 \)
- \( 61^2 = 3721 \)
- \( 7 \cdot 61 = 427 \)

Now, substituting these values back into the equation:
\[ M(7, 61) = 49 + 3721 + 427 \]
Calculating this step-by-step:
\[ 49 + 3721 = 3770 \]
\[ 3770 + 427 = 4197 \]
Thus, we have:
\[ M(7, 61) = 4197 \]

### Final Answer
The value of \( M(7, N(9, 4)) \) is \( 4197 \).

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