Question

use a suitable identity to get each of the following products
$1) \: (2a - 7)(2a - 7)$
$2) \: (x + 3)(x + 3)$
$3) \: (7a - 9b)(7a - 9b)$

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Answer 1

Answer:

A suitable identity to get each of the following products

[tex]1) \: (2a - 7)(2a - 7)

[tex]2) \: (x + 3)(x + 3)

[tex]3) \: (7a - 9b)(7a - 9b)

Answer 2

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Question\;:-}}}$

Here the concept of Algebraic Identities has been used. We see that the expressions are multiplied by themselves. So we can apply appropriate identity and then solve them.

Let's do it !!

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★ Identity Used :-

$\\\;\boxed{\sf{(a\;-\;b)^{2}\;=\;\bf{a^{2}\;-\;2ab\;+\;b^{2}}}}$

$\\\;\boxed{\sf{(a\;+\;b)^{2}\;=\;\bf{a^{2}\;+\;2ab\;+\;b^{2}}}}$

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★ Solution :-

1.) (2a - 7)(2a - 7)

2.) (x + 3)(x + 3)

3.) (7a - 9b)(7a - 9b)

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1.) (2a - 7)(2a - 7) ::

Here its, given that,

✒ (2a - 7)(2a - 7) = (2a - 7)²

Here, a = 2a and b = 7

Now applying the identity here, we get,

$\\\;\;\sf{:\rightarrow\;\;(a\;-\;b)^{2}\;=\;\bf{a^{2}\;-\;2ab\;+\;b^{2}}}$

$\\\;\;\sf{:\rightarrow\;\;(2a\;-\;7)^{2}\;=\;\bf{(2a)^{2}\;-\;2(2a)(7)\;+\;(7)^{2}}}$

$\\\;\;\bf{:\Longrightarrow\;\;(2a\;-\;7)^{2}\;=\;\bf{4a^{2}\;-\;28a\;+\;49}}$

$\\\;\underline{\boxed{\tt{(2a\;-\;7)(2a\;-\;7)\;=\;\bf{4a^{2}\;-\;28a\;+\;49}}}}$

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2.) (x + 3)(x + 3) ::

Here its given that,

✒ (x + 3)(x + 3) = (x + 3)²

Here, a = x and b = 3

Now applying the identity here, we get,

$\\\;\;\sf{:\rightarrow\;\;(a\;+\;b)^{2}\;=\;\bf{a^{2}\;+\;2ab\;+\;b^{2}}}$

$\\\;\;\sf{:\rightarrow\;\;(x\;+\;3)^{2}\;=\;\bf{(x)^{2}\;+\;2(x)(3)\;+\;(3)^{2}}}$

$\\\;\;\bf{:\Longrightarrow\;\;(x\;-\;3)^{2}\;=\;\bf{x^{2}\;+\;6x\;+\;9}}$

$\\\;\underline{\boxed{\tt{(x\;+\;3)(x\;+\;3)\;=\;\bf{x^{2}\;+\;6x\;+\;9}}}}$

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3.) (7a - 9b)(7a - 9b) ::

Here its given that,

✒ (7a - 9b)(7a - 9b) = (7a - 9b)²

Here, a = 7a and b = 9b

By applying the identity here, we get,

$\\\;\;\sf{:\rightarrow\;\;(a\;-\;b)^{2}\;=\;\bf{a^{2}\;-\;2ab\;+\;b^{2}}}$

$\\\;\;\sf{:\rightarrow\;\;(7a\;-\;9b)^{2}\;=\;\bf{(7a)^{2}\;-\;2(7a)(9b)\;+\;(9b)^{2}}}$

$\\\;\;\bf{:\Longrightarrow\;\;(2a\;-\;7)^{2}\;=\;\bf{49a^{2}\;-\;126ab\;+\;81b^{2}}}$

$\\\;\underline{\boxed{\tt{(7a\;-\;9b)(7a\;-\;9b)\;=\;\bf{49a^{2}\;-\;126ab\;+\;81b^{2}}}}}$

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★ More to know :-

$\\\;\sf{\leadsto\;\;a^{2}\;-\;b^{2}\;=\;(a\;+\;b)(a\;-\;b)}$

$\\\;\sf{\leadsto\;\;(a\;+\;b\;+\;c)^{2}\;=\;a^{2}\;+\;b^{2}\;+\;c^{2}\;+\;2ab\;+\;2bc\;+\;2ac}$

$\\\;\sf{\leadsto\;\;(x\;+\;a)(x\;+\;b)\;=\;x^{2}\;+\;(a\;+\;b)x\;+\;ab}$

$\\\;\sf{\leadsto\;\;(a\;+\;b)^{3}\;=\;a^{3}\;+\;b^{3}\;+\;3ab(a\;+\;b)}$

$\\\;\sf{\leadsto\;\;(a\;-\;b)^{3}\;=\;a^{3}\;-\;b^{3}\;-\;3ab(a\;-\;b)}$