a − b = 42 + 12 6 a-b=\sqrt{42+12\sqrt{6}}

Algebra Level 2

Given a − b = 42 + 12 6 , b − c = 42 − 12 6 , a-b=\sqrt{42+12\sqrt{6}}, b-c=\sqrt{42-12\sqrt{6}}, what is the value of a 2 + b 2 + c 2 − a b − b c − c a ? a^2+b^2+c^2-ab-bc-ca?

118 118 112 112 116 116 114 114

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Mahindra Jain by Mahindra Jain

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5 solutions

Chew-Seong Cheong
Jan 25, 2015

a − b = 42 + 12 6 a-b=\sqrt{42+12\sqrt{6}}

⇒ ( a − b ) 2 = 42 + 12 6 = 6 2 − 2 ( 6 ) ( − 6 ) + ( − 6 ) 2 = ( 6 + 6 ) 2 \Rightarrow (a-b)^2 = 42+12\sqrt{6} = 6^2 - 2(6)(-\sqrt{6})+(-\sqrt{6})^2 = (6+\sqrt{6})^2

⇒ a = 6 b = − 6 \Rightarrow a = 6 \quad b = -\sqrt{6}

Similarly, 42 − 12 6 = ( 6 − 6 ) 2 ⇒ c = − 6 42-12\sqrt{6} = (6-\sqrt{6})^2 \quad \Rightarrow c = -6

Therefore.

a 2 + b 2 + c 2 − a b − b c − c a = 36 + 6 + 36 + 6 6 − 6 6 + 36 = 114 a^2+b^2+c^2-ab-bc-ca = 36+6+36+6\sqrt{6}-6\sqrt{6}+36 = \boxed{114}

3 Helpful 0 Interesting 1 Brilliant 0 Confused
Mas Mus
Apr 13, 2014

a − b = 42 + 12 6 = 6 + 6 . . . . . ( 1 ) a 2 + b 2 − 2 a b = 42 + 12 6 . . . . . ( 2 ) b − c = 42 − 12 6 = 6 − 6 . . . . . ( 3 ) b 2 + c 2 − 2 b c = 42 − 12 6 . . . . . ( 4 ) % M a t h T y p e ! M T E F ! 2 ! 1 ! + − % f e a a g K a r t 1 e v 2 a a a t C v A U f e B S j u y Z L 2 y d 9 g z L b v y N v 2 C a e r b u L w B L n % h i o v 2 D G i 1 B T f M B a e X a t L x B I 9 g B a e r b d 9 w D Y L w z Y b I t L D h a r q q t u b s r % 4 r N C H b G e a G q i V u 0 J e 9 s q q r p e p C 0 x b b L 8 F 4 r q q r F f p e e a 0 x e 9 L q − J c 9 % v q a q p e p m 0 x b b a 9 p w e 9 Q 8 f s 0 − y q a q p e p a e 9 p g 0 F i r p e p e K k F r 0 x f r − x % f r − x b 9 a d b a q a a e G a c i G a a i a a b e q a a m a a b a a b a a G c e a q a b e a a c a W G H b % G a e y O e I 0 I a a m O y a i a b g 2 d a 9 m a a k a a a b a G a a G i n a i a a i k d a c q G H R a W k % c a a I X a G a a G O m a m a a k a a a b a G a a G O n a a W c b e a a a e q a a O G a e y y p a 0 J a a G % O n a i a b g U c a R m a a k a a a b a G a a G O n a a W c b e a a k i a a c 6 c a c a G G U a G a a i O l % a i a a c 6 c a c a G G U a G a a i i k a i a a i g d a c a G G P a a a b a G a a m y y a m a a C a a a l e % q a b a G a a G O m a a a a k i a b g U c a R i a a d k g a d a a h a a W c b e q a a i a a i k d a a a G c % c q G H s i s l c a a I Y a G a a m y y a i a a d k g a c q G H 9 a q p c a a I 0 a G a a G O m a i a b g U % c a R i a a i g d a c a a I Y a W a a O a a a e a a c a a I 2 a a a l e q a a O G a a i O l a i a a c 6 c a % c a G G U a G a a i O l a i a a c 6 c a c a G G O a G a a G O m a i a a c M c a a e a a c a W G I b G a e y % O e I 0 I a a m 4 y a i a b g 2 d a 9 m a a k a a a b a G a a G i n a i a a i k d a c q G H s i s l c a a I % X a G a a G O m a m a a k a a a b a G a a G O n a a W c b e a a a e q a a O G a e y y p a 0 J a a G O n a i % a b g k H i T m a a k a a a b a G a a G O n a a W c b e a a k i a a c 6 c a c a G G U a G a a i O l a i a a % c 6 c a c a G G U a G a a i i k a i a a i o d a c a G G P a a a b a G a a m O y a m a a C a a a l e q a b a % G a a G O m a a a a k i a b g U c a R i a a d o g a d a a h a a W c b e q a a i a a i k d a a a G c c q G H % s i s l c a a I Y a G a a m O y a i a a d o g a c q G H 9 a q p c a a I 0 a G a a G O m a i a b g k H i T i % a a i g d a c a a I Y a W a a O a a a e a a c a a I 2 a a a l e q a a O G a a i O l a i a a c 6 c a c a G G % U a G a a i O l a i a a c 6 c a c a G G O a G a a G i n a i a a c M c a a a a a ! 8020 ! \begin{array}{l}a - b = \sqrt {42 + 12\sqrt 6 } = 6 + \sqrt 6 .....(1)\\{a^2} + {b^2} - 2ab = 42 + 12\sqrt 6 .....(2)\\b - c = \sqrt {42 - 12\sqrt 6 } = 6 - \sqrt 6 .....(3)\\{b^2} + {c^2} - 2bc = 42 - 12\sqrt 6 .....(4)\end{array} \% MathType!MTEF!2!1!+- \% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn \% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr \% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 \% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x \% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb \% GaeyOeI0IaamOyaiabg2da9maakaaabaGaaGinaiaaikdacqGHRaWk \% caaIXaGaaGOmamaakaaabaGaaGOnaaWcbeaaaeqaaOGaeyypa0JaaG \% OnaiabgUcaRmaakaaabaGaaGOnaaWcbeaakiaac6cacaGGUaGaaiOl \% aiaac6cacaGGUaGaaiikaiaaigdacaGGPaaabaGaamyyamaaCaaale \% qabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaikdaaaGc \% cqGHsislcaaIYaGaamyyaiaadkgacqGH9aqpcaaI0aGaaGOmaiabgU \% caRiaaigdacaaIYaWaaOaaaeaacaaI2aaaleqaaOGaaiOlaiaac6ca \% caGGUaGaaiOlaiaac6cacaGGOaGaaGOmaiaacMcaaeaacaWGIbGaey \% OeI0Iaam4yaiabg2da9maakaaabaGaaGinaiaaikdacqGHsislcaaI \% XaGaaGOmamaakaaabaGaaGOnaaWcbeaaaeqaaOGaeyypa0JaaGOnai \% abgkHiTmaakaaabaGaaGOnaaWcbeaakiaac6cacaGGUaGaaiOlaiaa \% c6cacaGGUaGaaiikaiaaiodacaGGPaaabaGaamOyamaaCaaaleqaba \% GaaGOmaaaakiabgUcaRiaadogadaahaaWcbeqaaiaaikdaaaGccqGH \% sislcaaIYaGaamOyaiaadogacqGH9aqpcaaI0aGaaGOmaiabgkHiTi \% aaigdacaaIYaWaaOaaaeaacaaI2aaaleqaaOGaaiOlaiaac6cacaGG \% UaGaaiOlaiaac6cacaGGOaGaaGinaiaacMcaaaaa!8020! Add (1) and (3)

a − c = 12 a 2 + c 2 − 2 a c = 144..... ( 5 ) % M a t h T y p e ! M T E F ! 2 ! 1 ! + − % f e a a g K a r t 1 e v 2 a a a t C v A U f e B S j u y Z L 2 y d 9 g z L b v y N v 2 C a e r b u L w B L n % h i o v 2 D G i 1 B T f M B a e X a t L x B I 9 g B a e r b d 9 w D Y L w z Y b I t L D h a r q q t u b s r % 4 r N C H b G e a G q i V u 0 J e 9 s q q r p e p C 0 x b b L 8 F 4 r q q r F f p e e a 0 x e 9 L q − J c 9 % v q a q p e p m 0 x b b a 9 p w e 9 Q 8 f s 0 − y q a q p e p a e 9 p g 0 F i r p e p e K k F r 0 x f r − x % f r − x b 9 a d b a q a a e G a c i G a a i a a b e q a a m a a b a a b a a G c e a q a b e a a c a W G H b % G a e y O e I 0 I a a m 4 y a i a b g 2 d a 9 i a a i g d a c a a I Y a a a b a G a a m y y a m a a C a a a % l e q a b a G a a G O m a a a a k i a b g U c a R i a a d o g a d a a h a a W c b e q a a i a a i k d a a a % G c c q G H s i s l c a a I Y a G a a m y y a i a a d o g a c q G H 9 a q p c a a I X a G a a G i n a i a a % i s d a c a G G U a G a a i O l a i a a c 6 c a c a G G U a G a a i O l a i a a c I c a c a a I 1 a G a a i % y k a a a a a a ! 4 C 11 ! \begin{array}{l}a - c = 12\\{a^2} + {c^2} - 2ac = 144.....(5)\end{array} \% MathType!MTEF!2!1!+- \% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn \% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr \% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 \% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x \% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb \% GaeyOeI0Iaam4yaiabg2da9iaaigdacaaIYaaabaGaamyyamaaCaaa \% leqabaGaaGOmaaaakiabgUcaRiaadogadaahaaWcbeqaaiaaikdaaa \% GccqGHsislcaaIYaGaamyyaiaadogacqGH9aqpcaaIXaGaaGinaiaa \% isdacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaacIcacaaI1aGaai \% ykaaaaaa!4C11! Add (2), (4), (5)

2 ( a 2 + b 2 + c 2 − a b − a c − b c ) = 228 a 2 + b 2 + c 2 − a b − a c − b c = 114 % M a t h T y p e ! M T E F ! 2 ! 1 ! + − % f e a a g K a r t 1 e v 2 a a a t C v A U f e B S j u y Z L 2 y d 9 g z L b v y N v 2 C a e r b u L w B L n % h i o v 2 D G i 1 B T f M B a e X a t L x B I 9 g B a e r b d 9 w D Y L w z Y b I t L D h a r q q t u b s r % 4 r N C H b G e a G q i V u 0 J e 9 s q q r p e p C 0 x b b L 8 F 4 r q q r F f p e e a 0 x e 9 L q − J c 9 % v q a q p e p m 0 x b b a 9 p w e 9 Q 8 f s 0 − y q a q p e p a e 9 p g 0 F i r p e p e K k F r 0 x f r − x % f r − x b 9 a d b a q a a e G a c i G a a i a a b e q a a m a a b a a b a a G c e a q a b e a a c a a I Y a % G a a i i k a i a a d g g a d a a h a a W c b e q a a i a a i k d a a a G c c q G H R a W k c a W G I b W a % a W b a a S q a b e a a c a a I Y a a a a O G a e y 4 k a S I a a m 4 y a m a a C a a a l e q a b a G a a G % O m a a a a k i a b g k H i T i a a d g g a c a W G I b G a e y O e I 0 I a a m y y a i a a d o g a c q G H % s i s l c a W G I b G a a m 4 y a i a a c M c a c q G H 9 a q p c a a I Y a G a a G O m a i a a i I d a a e % a a c a W G H b W a a W b a a S q a b e a a c a a I Y a a a a O G a e y 4 k a S I a a m O y a m a a C a a a % l e q a b a G a a G O m a a a a k i a b g U c a R i a a d o g a d a a h a a W c b e q a a i a a i k d a a a % G c c q G H s i s l c a W G H b G a a m O y a i a b g k H i T i a a d g g a c a W G J b G a e y O e I 0 I a % a m O y a i a a d o g a c q G H 9 a q p c a a I X a G a a G y m a i a a i s d a a a a a ! 5 D 92 ! \begin{array}{l}2({a^2} + {b^2} + {c^2} - ab - ac - bc) = 228\\{a^2} + {b^2} + {c^2} - ab - ac - bc = 114\end{array} \% MathType!MTEF!2!1!+- \% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn \% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr \% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 \% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x \% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIYa \% GaaiikaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGIbWa \% aWbaaSqabeaacaaIYaaaaOGaey4kaSIaam4yamaaCaaaleqabaGaaG \% OmaaaakiabgkHiTiaadggacaWGIbGaeyOeI0IaamyyaiaadogacqGH \% sislcaWGIbGaam4yaiaacMcacqGH9aqpcaaIYaGaaGOmaiaaiIdaae \% aacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaa \% leqabaGaaGOmaaaakiabgUcaRiaadogadaahaaWcbeqaaiaaikdaaa \% GccqGHsislcaWGHbGaamOyaiabgkHiTiaadggacaWGJbGaeyOeI0Ia \% amOyaiaadogacqGH9aqpcaaIXaGaaGymaiaaisdaaaaa!5D92!

3 Helpful 0 Interesting 0 Brilliant 0 Confused

114:- (a-b)^+(b-c)^+(c-a)^ = 2a^+2b^+2c^-2ab-2bc-2ca =228. Hence a^+b^+c^-ab-bc-ca = 114.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Rajesh Siriset
Mar 24, 2014

(a-b)(b-c) = 30 --> (1) and (a-b)^2 + (b-c)^2 = 84 --> (2) now add(1) & (2) we get a^2+b^2+c^2-ab-bc-ac = 30+84 = 114

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Arun Kasi
Mar 22, 2014

Given:
a-b=√(42+12√6)

b-c=√(42-12√6)

then c-a= -(b-c)-(a-b)

=-(√(42+12√6) +√(42-12√6) ))

a^2+b^2+c^2-ab-bc-ca=((a-b)^2+(b-c)^2+(c-a)^2)/2

=(42+12√6+42-12√6+42-12√6+42+12√6+60)/2

=114

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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