Question

If a + b + c = 8, ab + bc + ca = 21, then find the value of a2 + b2 + c2 .​

Answer 1

Step-by-step explanation:

Given:

ab + bc + ca = 8 and a2 + b2 + c2 = 20

Formula used:

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

Calculation:

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

⇒ (a + b + c)2 = 20 + 2 × 8 = 36

⇒ (a + b + c) = 6

Now,

1/2 × (a + b + c)[(a – b)2 + (b – c)2 + (c – a)2]

⇒ 1/2(a + b + c)(a2 + b2 – 2ab + b2 + c2 – 2bc + c2 + a2 – 2ac)

⇒ 1/2(a + b + c)[2(a2 + b2 + c2) – 2(ab + bc + ac)]

Now substituting the values,

⇒ (a + b + c)[(a2 + b2 + c2) – (ab + bc + ac)]

⇒ 6(20 – 8)

⇒ 6 × 12 = 72

∴ The required answer is 72.