Question

A+b+c=15 and ab+BC+ca=75 find 3a^2+3b^2+3c^2

Answer 1

${(a + b + c)}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2(ab + bc + ca) \\ \\ \Rightarrow {15}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2(75) \\ \\ \Rightarrow225 = {a}^{2} + {b}^{2} + {c}^{2} + 150 \\ \\ \Rightarrow{a}^{2} + {b}^{2} + {c}^{2} = 225 - 150 \\ \\ \Rightarrow {a}^{2} + {b}^{2} + {c}^{2} =75 \\ \\ \Rightarrow 3( {a}^{2} + {b}^{2} + {c}^{2}) = 3 \times 75 \\ \\ \Rightarrow \boxed{3{a}^{2} + 3{b}^{2} + 3 {c}^{2} =225}$

Answer 2

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${(a + b + c)}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2(ab + bc + ca) \\ \\ \Rightarrow {15}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2(75) \\ \\ \Rightarrow225 = {a}^{2} + {b}^{2} + {c}^{2} + 150 \\ \\ \Rightarrow{a}^{2} + {b}^{2} + {c}^{2} = 225 - 150 \\ \\ \Rightarrow {a}^{2} + {b}^{2} + {c}^{2} =75 \\ \\ \Rightarrow 3( {a}^{2} + {b}^{2} + {c}^{2}) = 3 \times 75 \\ \\ \Rightarrow \boxed{3{a}^{2} + 3{b}^{2} + 3 {c}^{2} =225}$

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