evaluate : (2a + 3b + 4c) power 2 + (2a – 3b + 4c) power 2 + (2a + 3b – 4c) power 2 About the author Josephine
[tex] \Large {\underline { \sf {Answer :}}}[/tex] 12a² + 27b² + 48c² + 12ab – 24bc + 16ac [tex] \Large {\underline { \sf {Clarification :}}}[/tex] Here, we are asked to find the value of the given expression : (2a + 3b + 4c)² + (2a – 3b + 4c)² + (2a + 3b – 4c)² Required Identity : (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) Required Steps : • Step 1 : We’ll solve each of the expression separately by using the above identity. • Step 2 : After getting the value of each expression, we’ll find their sum in order to evaluate the given expression. [tex] \Large {\underline { \sf {Explication \; of \; Steps :}}}[/tex] ⇒ (2a + 3b + 4c)² + (2a – 3b + 4c)² + (2a + 3b – 4c)² Expression 1 : (2a + 3b + 4c)² Expression 2 : (2a – 3b + 4c)² Expression 3 : (2a + 3b – 4c)² Finding the value of expression 1 : [tex]\longrightarrow \sf{ (2a + 3b + 4c)^2} [/tex] By using identity, (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) → (2a)² + (3b)² + (4c)² + 2{ (2a × 3b) + (3b × 4c) + (4c × 2a) } → 4a² + 9b² + 16c² + 2{ (6ab) + (12bc) + (8ca) } → 4a² + 9b² + 16c² + 2(6ab) + 2(12bc) + 2(8ca) → 4a² + 9b² + 16c² + 12ab + 24bc + 16ac Finding the value of expression 2 : [tex]\longrightarrow \sf{ (2a – 3b + 4c)^2} [/tex] We can also write it as, → {2a +(-3b) + 4c}² By using identity, (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) → (2a)² + (-3b)² + (4c)² + 2[ {2a × (-3b)} + {(-3b) × 4c} + (4c × 2a) ] → 4a² + 9b² + 16c² + 2[ {-6ab} + {-12bc} + (8ca) ] → 4a² + 9b² + 16c² + 2{-6ab} + 2{-12bc} + 2(8ca) → 4a² + 9b² + 16c² – 12ab – 24bc + 16ca Finding the value of expression 3 : [tex]\longrightarrow \sf{ (2a + 3b – 4c)^2} [/tex] We can also write it as, → {2a + 3b + (-4c)}² By using identity, (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) → (2a)² + (3b)² + (-4c)² + 2[ (2a × 3b) + {3b × (-4c)} + {(-4c) × 2a} ] → 4a² + 9b² + 16c² + 2[ (6ab) + {-12bc} + {-8ca} ] → 4a² + 9b² + 16c² + 2(6ab) + 2{-12bc} + 2(-8ca) → 4a² + 9b² + 16c² + 12ab – 24bc – 16ca Combining all the three expressions : → (4a² + 9b² + 16c² + 12ab + 24bc + 16ac) + (4a² + 9b² + 16c² – 12ab – 24bc + 16ca) + (4a² + 9b² + 16c² + 12ab – 24bc – 16ca) → 4a² + 9b² + 16c² + 12ab + 24bc + 16ac + 4a² + 9b² + 16c² – 12ab – 24bc + 16ca + 4a² + 9b² + 16c² + 12ab – 24bc – 16ca → 4a² + 4a² + 4a² + 9b² + 9b² + 9b² + 16c² + 16c² + 16c² + 12ab + 12ab – 12ab + 24bc – 24bc – 24bc + 16ac – 16ac + 16ac → 12a² + 27b² + 48c² + 24ab – 12ab – 24bc + 16ac → 12a² + 27b² + 48c² + 12ab – 24bc + 16ac Therefore, 12a² + 27b² + 48c² + 12ab – 24bc + 16ac is the required answer. Reply
[tex] \Large {\underline { \sf {Answer :}}}[/tex]
12a² + 27b² + 48c² + 12ab – 24bc + 16ac
[tex] \Large {\underline { \sf {Clarification :}}}[/tex]
Here, we are asked to find the value of the given expression :
Required Identity :
Required Steps :
• Step 1 : We’ll solve each of the expression separately by using the above identity.
• Step 2 : After getting the value of each expression, we’ll find their sum in order to evaluate the given expression.
[tex] \Large {\underline { \sf {Explication \; of \; Steps :}}}[/tex]
⇒ (2a + 3b + 4c)² + (2a – 3b + 4c)² + (2a + 3b – 4c)²
Finding the value of expression 1 :
[tex]\longrightarrow \sf{ (2a + 3b + 4c)^2} [/tex]
By using identity,
→ (2a)² + (3b)² + (4c)² + 2{ (2a × 3b) + (3b × 4c) + (4c × 2a) }
→ 4a² + 9b² + 16c² + 2{ (6ab) + (12bc) + (8ca) }
→ 4a² + 9b² + 16c² + 2(6ab) + 2(12bc) + 2(8ca)
→ 4a² + 9b² + 16c² + 12ab + 24bc + 16ac
Finding the value of expression 2 :
[tex]\longrightarrow \sf{ (2a – 3b + 4c)^2} [/tex]
We can also write it as,
→ {2a +(-3b) + 4c}²
By using identity,
→ (2a)² + (-3b)² + (4c)² + 2[ {2a × (-3b)} + {(-3b) × 4c} + (4c × 2a) ]
→ 4a² + 9b² + 16c² + 2[ {-6ab} + {-12bc} + (8ca) ]
→ 4a² + 9b² + 16c² + 2{-6ab} + 2{-12bc} + 2(8ca)
→ 4a² + 9b² + 16c² – 12ab – 24bc + 16ca
Finding the value of expression 3 :
[tex]\longrightarrow \sf{ (2a + 3b – 4c)^2} [/tex]
We can also write it as,
→ {2a + 3b + (-4c)}²
By using identity,
→ (2a)² + (3b)² + (-4c)² + 2[ (2a × 3b) + {3b × (-4c)} + {(-4c) × 2a} ]
→ 4a² + 9b² + 16c² + 2[ (6ab) + {-12bc} + {-8ca} ]
→ 4a² + 9b² + 16c² + 2(6ab) + 2{-12bc} + 2(-8ca)
→ 4a² + 9b² + 16c² + 12ab – 24bc – 16ca
Combining all the three expressions :
→ (4a² + 9b² + 16c² + 12ab + 24bc + 16ac) + (4a² + 9b² + 16c² – 12ab – 24bc + 16ca) + (4a² + 9b² + 16c² + 12ab – 24bc – 16ca)
→ 4a² + 9b² + 16c² + 12ab + 24bc + 16ac + 4a² + 9b² + 16c² – 12ab – 24bc + 16ca + 4a² + 9b² + 16c² + 12ab – 24bc – 16ca
→ 4a² + 4a² + 4a² + 9b² + 9b² + 9b² + 16c² + 16c² + 16c² + 12ab + 12ab – 12ab + 24bc – 24bc – 24bc + 16ac – 16ac + 16ac
→ 12a² + 27b² + 48c² + 24ab – 12ab – 24bc + 16ac
→ 12a² + 27b² + 48c² + 12ab – 24bc + 16ac
Therefore, 12a² + 27b² + 48c² + 12ab – 24bc + 16ac is the required answer.
hope it help more in future so