If `ab + bc + ca = 8` and `a+b+c = 12` then `(a^(2) + b^(2) + c^(2))` is equal to :

Options

A. 160
B. 128
C. 134
D. 144

Answer

To solve the problem, we will use the identity that relates the squares of the sums of variables to the sum of their squares and their products. The identity we will use is:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

Given:
1. \( ab + bc + ca = 8 \)
2. \( a + b + c = 12 \)

We need to find \( a^2 + b^2 + c^2 \).

### Step 1: Expand the identity
Using the identity, we can expand \( (a + b + c)^2 \):

\[ (a + b + c)^2 = 12^2 = 144 \]

### Step 2: Substitute the known values
Now, substitute the known values into the identity:

\[ 144 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

Substituting \( ab + bc + ca = 8 \):

\[ 144 = a^2 + b^2 + c^2 + 2 \times 8 \]

### Step 3: Simplify the equation
Calculate \( 2 \times 8 \):

\[ 2 \times 8 = 16 \]

Now substitute this back into the equation:

\[ 144 = a^2 + b^2 + c^2 + 16 \]

### Step 4: Isolate \( a^2 + b^2 + c^2 \)
To find \( a^2 + b^2 + c^2 \), subtract 16 from both sides:

\[ a^2 + b^2 + c^2 = 144 - 16 \]

### Step 5: Perform the final calculation
Calculate \( 144 - 16 \):

\[ a^2 + b^2 + c^2 = 128 \]

Thus, the value of \( a^2 + b^2 + c^2 \) is \( 128 \).

### Final Answer
\[ \boxed{128} \]

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