a 2 + b 2 + c 2 a^2+b^2+c^2

Algebra Level 2

Given that a + b + c = 8 a+b+c=8 and a b + b c + c a = − 2 ab+bc+ca=-2 , what is a 2 + b 2 + c 2 a^2+b^2+c^2 ?

72 64 68 66

Kano Boom by Kano Boom , 16, Australia

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2 solutions

Hi Bye
Oct 30, 2019

Note that ( a + b + c ) 2 = ( a 2 + b 2 + c 2 ) + 2 ( a b + b c + a c ) . (a+b+c)^2=\left(a^2+b^2+c^2\right)+2(ab+bc+ac). Since we are given the value of a + b + c a+b+c and a b + b c + c a , ab+bc+ca, we easily find that ( a 2 + b 2 + c 2 ) = 8 2 − ( − 4 ) = 68 . \left(a^2+b^2+c^2\right)=8^2-(-4)=\boxed{68}.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Henry U
Oct 30, 2019

( a + b + c ) 2 = a a + a b + a c + b a + b b + b c + c a + c b + c c ⇔ ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 ( a b + a c + b c ) ⇔ ( 8 ) 2 = a 2 + b 2 + c 2 + 2 ( − 2 ) ⇔ a 2 + b 2 + c 2 = 8 2 + 2 ⋅ 2 \begin{aligned} && (a+b+c)^2 &= aa+ab+ac+ba+bb+bc+ca+cb+cc \\ \Leftrightarrow && ({\color{#3D99F6}a+b+c})^2 &= a^2+b^2+c^2+2({\color{#D61F06}ab+ac+bc}) \\ \Leftrightarrow && ({\color{#3D99F6}8})^2 &= a^2+b^2+c^2+2({\color{#D61F06}-2}) \\ \Leftrightarrow && a^2+b^2+c^2 &= 8^2 + 2 \cdot 2 \end{aligned}

a 2 + b 2 + c 2 = 68 \boxed{a^2+b^2+c^2 = 68}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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