If `a+b+c=7 and a^(3)+b^(3)+c^(3)-3abc=175`, then what is the value of `(ab+bc+ca)`?

Options

A. 8
B. 7
C. 6
D. 9

Answer

To solve the problem step by step, we will use the provided equations and manipulate them to find the value of \( ab + bc + ca \).

### Step 1: Write down the given equations
We have two equations:
1. \( a + b + c = 7 \)
2. \( a^3 + b^3 + c^3 - 3abc = 175 \)

### Step 2: Use the identity for the sum of cubes
We can use the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
Substituting \( a + b + c = 7 \) into the equation gives us:
\[ a^3 + b^3 + c^3 - 3abc = 7(a^2 + b^2 + c^2 - ab - ac - bc) \]

### Step 3: Substitute the second equation
From the second equation, we know:
\[ 175 = 7(a^2 + b^2 + c^2 - ab - ac - bc) \]
Dividing both sides by 7:
\[ a^2 + b^2 + c^2 - ab - ac - bc = 25 \]

### Step 4: Use the square of the sum of variables
We know that:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
Substituting \( a + b + c = 7 \):
\[ 7^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
This simplifies to:
\[ 49 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

### Step 5: Express \( a^2 + b^2 + c^2 \)
From our previous result, we can express \( a^2 + b^2 + c^2 \) as:
\[ a^2 + b^2 + c^2 = 25 + ab + ac + bc \]

### Step 6: Substitute \( a^2 + b^2 + c^2 \) into the square equation
Now substitute \( a^2 + b^2 + c^2 \) into the equation:
\[ 49 = (25 + ab + ac + bc) + 2(ab + ac + bc) \]
This simplifies to:
\[ 49 = 25 + 3(ab + ac + bc) \]

### Step 7: Solve for \( ab + ac + bc \)
Rearranging gives:
\[ 49 - 25 = 3(ab + ac + bc) \]
\[ 24 = 3(ab + ac + bc) \]
Dividing both sides by 3:
\[ ab + ac + bc = 8 \]

### Final Answer
Thus, the value of \( ab + ac + bc \) is \( \boxed{8} \).