Question

If a^2+ b^2 = 9 and ab = 4.
Find the value of 3(a + b)^2 - 2(a - b)^2​

Answer 1

Answer :

The value of 3 ( a + b )² - 2 ( a - b )² is 49

Step-by-step explanation :

To Find,

• The value of 3 ( a + b )² - 2 ( a - b )²

Solution,

Given that,

• a² + b² = 9
• ab = 4

Therefore,

⟹ 3 ( a + b )² - 2 ( a - b )²

⟹ 3 ( a² + b² + 2ab ) - 2 ( a² + b² - 2ab )

⟹ 3 ( 9 + 2 × 4 ) - 2 ( 9 - 2 × 4 )

⟹ 3 ( 9 + 8 ) - 2 ( 9 - 8 )

⟹ 3 ( 17 ) - 2 ( 1 )

⟹ 3 × 17 - 2 × 1

⟹ 51 - 2 × 1

⟹ 51 - 2

⟹ 49 ★

Now, Verification

3 ( a + b )² - 2 ( a - b )² = 49

By putting the values,

⟹ 3 ( a + b )² - 2 ( a - b )² = 49

⟹ 3 ( a² + b² + 2ab ) - 2 ( a² + b² - 2ab ) = 49

⟹ 3 ( 9 + 2 × 4 ) - 2 ( 9 - 2 × 4 ) = 49

⟹ 3 ( 9 + 8 ) - 2 ( 9 - 8 ) = 49

⟹ 3 ( 17 ) - 2 ( 1 ) = 49

⟹ 3 × 17 - 2 × 1 = 49

⟹ 51 - 2 = 49

⟹ 49 = 49

L.H.S = R.H.S

Hence, Verified !

Answer 2

$\\ \\ \large\underline{ \underline{ \sf{ \red{given:} }}}$

• a² + b² = 9

• ab = 4

$\\ \\ \large\underline{ \underline{ \sf{ \red{ to \: find:} }}}$

• 3 ( a + b )² - 2 ( a - b )²

$\\ \\ \large\underline{ \underline{ \sf{ \red{solution:} }}}$

$\implies \sf \: 3( {a + b)}^{2} - 2( {a - b)}^{2} \\ \\ \\ \bigstar\boxed{ \bf \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy } \\ \\ \bigstar\boxed{ \bf \:( {x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy } \\ \\ \\ \implies \sf \: 3( {a}^{2} + {b}^{2} + 2ab) - 2( {a}^{2} + {b}^{2} - 2ab)$

We know ,

• a² + b² = 9

• ab = 4

Putting values ,

$\\ \implies \sf \: 3(9 + 2(4)) - 2(9 - 2(4)) \\ \\ \\ \implies \sf \: 3(9 + 8) - 2(9 - 8) \\ \\ \\ \implies\sf \: 3(17) - 2(1) \\ \\ \\ \implies\sf \: 51 - 2 \\ \\ \\ \sf \: \implies \blue{ 49}$

Hence , required answer is 49.

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More identities :-

• ( a + b ) ( a - b ) = a² - b²

• ( a + x ) ( a + y ) = a² + ( x + y )a + xy

• ( a + b )³ = a³ + 3a²b + 3ab² + b³