Question

If $a + b + c = 12$ and $a^2 + b^2 + c^2 = 50$, what is the value of $ab + bc + ca$?

• A. 47
• B. 48
• C. 49
• D. 50

Hint:

Use the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$ to find the sum of pairwise products quickly.

Answer 1

The Correct Option is A

- Step 1: Use the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$.
- Step 2: Given $a + b + c = 12$ and $a^2 + b^2 + c^2 = 50$, substitute: $12^2 = 50 + 2(ab + bc + ca)$.
- Step 3: Calculate: $144 = 50 + 2(ab + bc + ca)$.
- Step 4: Solve: $2(ab + bc + ca) = 144 - 50 = 94$, so $ab + bc + ca = \dfrac{94}{2} = 47$.
- Step 5: Verify: If $ab + bc + ca = 47$, then $2 \times 47 + 50 = 94 + 50 = 144$, which matches $12^2$.
- Step 6: Check options: Option (a) is 47, which matches.