Question

evaluate : (2a + 3b + 4c) power 2 + (2a - 3b + 4c) power 2 + (2a + 3b - 4c) power 2​

Answer 1

$\Large {\underline { \sf {Answer :}}}$

12a² + 27b² + 48c² + 12ab - 24bc + 16ac

$\Large {\underline { \sf {Clarification :}}}$

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.

$\Large {\underline { \sf {Explication \; of \; Steps :}}}$

⇒ (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 :

$\longrightarrow \sf{ (2a + 3b + 4c)^2}$

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 + 16ac

Finding the value of expression 2 :

$\longrightarrow \sf{ (2a - 3b + 4c)^2}$

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 :

$\longrightarrow \sf{ (2a + 3b - 4c)^2}$

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.

Answer 2

hope it help more in future so