If a + b + c = 9 and ab + bc + ca = 15, find: a² + b² + c².

Using the formula,

[∵(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx]

So,

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

⇒ (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Putting the value a + b + c = 9 and ab + bc + ca = 15, we get

⇒ 9² = a² + b² + c² + 2 x 15

⇒ 81 = a² + b² + c² + 30

⇒ a² + b² + c² = 81 - 30

⇒ a² + b² + c² = 51

Hence, the value of a² + b² + c² is 51.