Question

If a^2 + b^2 + c^2 = 5 and ab + bc + ca = 10, find the value of a+ b + c.

Answer 1

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

= 5 + 2 × 10 = 5 + 20 = 25

= a + b + c = ±√25

= ±5

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Answer 2

Given, a^2 + b^2 + c^2 = 5   ...( i )

            ab + bc + ca = 10      ...( ii )

Multiply by 2 on both sides of ( ii ),

⇒ 2( ab + bc + ca ) = 2( 10 )

⇒ 2( ab + bc + ca ) = 20      ...( iii )

Then, adding ( i ) and ( ii ),

⇒ a^2 + b^2 + c^2 + 2( ab + bc + ca ) = 5 + 20

⇒ a^2 + b^2 + c^2 + 2( ab + bc + ca ) = 25

We know that the value of a^2 + b^2 + c^2 + 2( ab + bc + ca ) in factorized form is ( a + b + c )^2.

∴ ( a + b + c )^2 = 25

⇒ ( a + b + c )^2 = ( 5 )^2 or ( - 5 )^2

⇒ ( a + b + c ) = 5 or - 5

Therefore the value of a + b + c is 5 or - 5.

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