If `a+b=7` and `ab=12` find the value of `a^(2)-ab+b^(2)`.

Options

A. `13`
B. `37`
C. `15`
D. `50`

Answer

To find the value of \( a^2 - ab + b^2 \) given that \( a + b = 7 \) and \( ab = 12 \), we can use the following steps:

### Step 1: Use the identity for \( a^2 + b^2 \)
We know that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting the values of \( a + b \) and \( ab \):
\[ a^2 + b^2 = (7)^2 - 2(12) \]

### Step 2: Calculate \( (a + b)^2 \)
Calculating \( (7)^2 \):
\[ (7)^2 = 49 \]

### Step 3: Calculate \( 2ab \)
Calculating \( 2(12) \):
\[ 2(12) = 24 \]

### Step 4: Substitute back to find \( a^2 + b^2 \)
Now substituting back into the equation:
\[ a^2 + b^2 = 49 - 24 = 25 \]

### Step 5: Substitute into \( a^2 - ab + b^2 \)
Now we need to find \( a^2 - ab + b^2 \). We can express this as:
\[ a^2 - ab + b^2 = a^2 + b^2 - ab \]
Substituting the values we found:
\[ a^2 - ab + b^2 = 25 - 12 \]

### Step 6: Calculate the final value
Calculating:
\[ a^2 - ab + b^2 = 25 - 12 = 13 \]

Thus, the value of \( a^2 - ab + b^2 \) is \( \boxed{13} \).

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