Question

If a2 + b2+ c2= 29 and a+b+c = 9,find ab+bc+ca

Answer 1

Given that

a + b + c = 10 ...eq1\\ \\ {a}^{2} + {b}^{2} + {c}^{2} = 29 ...eq2\\ \\

To find

ab + bc + ca \\ \\

Square eq1 both side

{(a + b + c)}^{2} = {(10)}^{2} \\ \\ {a}^{2} + {b}^{2} + {c}^{2} + 2ab + 2bc + 2ca = 100 \\ \\

because we know that

{(x + y + z)}^{2} = {x}^{2} + {y}^{2} + {z}^{2} + 2xy + 2yz + 2xz \\ \\

put the value of eq2

29 + 2ab + 2bc + 2ca = 100 \\ \\ 2ab + 2bc + 2ca = 100 - 29 \\ \\ 2(ab + bc + ca) = 71 \\ \\ ab + bc + ca = \frac{71}{2} \\ \\ \\ ab + bc + ca = 35.5 \\ \\