If `a^(2) + b^(2)= 34 and ab= 12`, find:
`7(a-b)^(2) - 2(a +b)^(2)`

Answer

To solve the problem, we need to find the value of \( 7(a-b)^2 - 2(a+b)^2 \) given that \( a^2 + b^2 = 34 \) and \( ab = 12 \).

### Step 1: Find \( (a-b)^2 \) and \( (a+b)^2 \)

We can use the identities:
\[ (a-b)^2 = a^2 + b^2 - 2ab \]
\[ (a+b)^2 = a^2 + b^2 + 2ab \]

### Step 2: Substitute the known values

We know:
- \( a^2 + b^2 = 34 \)
- \( ab = 12 \)

Now, substitute these values into the formulas.

1. For \( (a-b)^2 \):
\[ (a-b)^2 = 34 - 2 \cdot 12 = 34 - 24 = 10 \]

2. For \( (a+b)^2 \):
\[ (a+b)^2 = 34 + 2 \cdot 12 = 34 + 24 = 58 \]

### Step 3: Substitute into the expression

Now we substitute \( (a-b)^2 \) and \( (a+b)^2 \) into the expression \( 7(a-b)^2 - 2(a+b)^2 \):
\[ 7(a-b)^2 - 2(a+b)^2 = 7 \cdot 10 - 2 \cdot 58 \]

### Step 4: Calculate the expression

Now we calculate:
\[ 7 \cdot 10 = 70 \]
\[ 2 \cdot 58 = 116 \]
Thus,
\[ 7(a-b)^2 - 2(a+b)^2 = 70 - 116 = -46 \]

### Final Answer

The value of \( 7(a-b)^2 - 2(a+b)^2 \) is \( -46 \).

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