If `a^(2)+b^(2)+c^(2)=29,ab+bc+ca=26anda,b,c inN`, then find `a+b+c`.

Options

A. 9
B. 6
C. 7
D. 10

Answer

To solve the problem step by step, we will use the given equations and an algebraic identity.

### Step 1: Write down the given equations.
We have:
1. \( a^2 + b^2 + c^2 = 29 \)
2. \( ab + bc + ca = 26 \)

### Step 2: Use the identity for the square of a sum.
We know that:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

### Step 3: Substitute the known values into the identity.
Substituting the values from the equations into the identity:
\[ (a + b + c)^2 = 29 + 2 \times 26 \]

### Step 4: Calculate the right-hand side.
First, calculate \( 2 \times 26 \):
\[ 2 \times 26 = 52 \]
Now, add this to 29:
\[ 29 + 52 = 81 \]

### Step 5: Set up the equation for \( (a + b + c)^2 \).
Now we have:
\[ (a + b + c)^2 = 81 \]

### Step 6: Take the square root of both sides.
To find \( a + b + c \), we take the square root:
\[ a + b + c = \sqrt{81} = 9 \]

### Step 7: Conclusion.
Thus, the value of \( a + b + c \) is:
\[ \boxed{9} \]

---