The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. Their sum is

Options

A. 20
B. 30
C. 40
D. 50

Answer

To solve the problem, we need to find the sum of three numbers \( a \), \( b \), and \( c \) given the following information:

1. The sum of their squares:
\[ a^2 + b^2 + c^2 = 138 \]

2. The sum of their products taken two at a time:
\[ ab + bc + ca = 131 \]

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

### Step-by-Step Solution:

**Step 1: 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 2: Substitute the known values into the identity.**
Substituting the values we have:
\[ (a + b + c)^2 = 138 + 2 \times 131 \]

**Step 3: Calculate the right-hand side.**
First, calculate \( 2 \times 131 \):
\[ 2 \times 131 = 262 \]
Now, add this to 138:
\[ 138 + 262 = 400 \]

**Step 4: Set up the equation.**
Now we have:
\[ (a + b + c)^2 = 400 \]

**Step 5: Take the square root.**
To find \( a + b + c \), take the square root of both sides:
\[ a + b + c = \sqrt{400} \]

**Step 6: Calculate the square root.**
The square root of 400 is:
\[ \sqrt{400} = 20 \]

Thus, the sum of the three numbers \( a + b + c \) is:
\[ \boxed{20} \]