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Mathematics

If a + b + c = 11 and a2 + b2 + c2 = 81, find: ab + bc + ca.

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Answer

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) = 11 and a2 + b2 + c2 = 81,

⇒ (11)2 = 81 + 2(ab + bc + ca)

⇒ 121 = 81 + 2(ab + bc + ca)

⇒ 2(ab + bc + ca) = 121 - 81

⇒ 2(ab + bc + ca) = 40

⇒ ab + bc + ca = 402\dfrac{40}{2}

⇒ ab + bc + ca = 20

Hence, the value of (ab + bc + ca) is 20.

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