If a + b + c = 20 and a² + b² + c² = 200, find the value of (a − b)² + (b − c)² + (c − a)².
Show answer & explanation
Correct answer: C
Explanation
a + b + c = 20
a² + b² + c² = 200
According to the formula:
(a − b)² + (b − c)² + (c − a)² = 2(a² + b² + c² − ab − bc − ca)
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
20² = 200 + 2(ab + bc + ca)
400 = 200 + 2(ab + bc + ca)
2(ab + bc + ca) = 200
ab + bc + ca = 100
= 2(200 − 100)
= 2 × 100
= 200