Which of the following identities is not applicable to simplify
$(4 x + 5 y)^{2}$ - $(5 x + 4 y)^{2}$ ?

$a^{2} + b^{2} = (a + b)^{2} - 2 a b$

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$a^{2} - b^{2} = (a + b) (a - b)$

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$(a - b)^{2} = a^{2} - 2 a b + b^{2}$

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$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

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Solution

The correct option is C
$(a - b)^{2} = a^{2} - 2 a b + b^{2}$

$(4 x + 5 y)^{2}$
= $(4 x)^{2} + 2 \times (4 x) \times (5 y) \times (5 y)^{2}$
[ Using the identity
$(a + b)^{2} = a^{2} + 2 a b + b^{2}$
or
$a^{2} + b^{2} = (a + b)^{2} - 2 a b$ ]
So, $(4 x + 5 y)^{2} = 16 x^{2} + 40 x y + 25 y^{2}$

Similarly, $(5 x + 4 y)^{2} = 25 x^{2} + 40 x y + 16 y^{2}$
Now,
$(4 x + 5 y)^{2} - (5 x + 4 y)^{2}$
= $(16 x^{2} + 40 x y + 25 y^{2})$ -
$(25 x^{2} + 40 x y + 16 y^{2}$ )
$= 16 x^{2} + 40 x y + 25 y^{2}$ -
$25 x^{2} - 40 x y - 16 y^{2}$
$= 9 y^{2} - 9 x^{2}$
OR
Using Identity $a^{2} - b^{2} = (a + b) (a - b)$ ,
$(4 x + 5 y)^{2} - (5 x + 4 y)^{2}$
$= [(4 x + 5 y) - (5 x + 4 y)]$ $\times [(4 x + 5 y) + (5 x + 4 y)]$
$= (y - x) (9 x + 9 y)$
$= 9 x y - 9 x^{2} + 9 y^{2} - 9 x y$
$= 9 y^{2} - 9 x^{2}$

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Similar questions

Which of the following identity is not applicable to simplify ${(4 x + 5 y)}^{2} - {(5 x + 4 y)}^{2}$ ?

Which of the following identity is applicable to simplify (4x + 5y)^2 - (5x + 4y)^2 ?

3 x^{2} - 5 y^{2} = 7

3 x y - 4 y^{2} = 2

2 x^{2} - 4 y^{2} - z^{2}, 8 x^{2} + 5 y^{2} + 4 z^{2}, 5 x^{2} - 5 y^{2} + 2 z^{2}

Which of the following identity can be used to factorize $x^{2} - 4 y^{2}$ ?

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