Question

compute the following by using the above identities
$99 {}^{2}$
$(100.1) {}^{2}$
$49 \times 51$


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Answer 1

$\large\underline{\sf{Solution-a}}$

$\rm \: {99}^{2}$

can be rewritten as

$\rm \: = \: {(100 - 1)}^{2}$

We know,

$\boxed{\rm{  \: {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} \: }}$

Here,

$\rm \: x = 100$

$\rm \: y = 1$

So, on substituting these values, in above identity, we get

$\rm \: = \: {(100)}^{2} - 2 \times 100 \times 1 + {(1)}^{2}$

$\rm \: = \: 10000 - 200 + 1$

$\rm \: = \: 9801$

Hence,

$\red{\rm\implies \:\boxed{\rm{  \: {99}^{2} = 9801 \: }}}$

$\large\underline{\sf{Solution-b}}$

$\rm \: {(100.1)}^{2}$

$\rm \: = \: {(100 + 0.1)}^{2}$

We know,

$\boxed{\rm{  \: {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} \: }}$

So, here

$\rm \: x = 100$

$\rm \: y = 0.1$

So, on substituting the values in above identity, we get

$\rm \: = \: {(100)}^{2} + 2 \times 100 \times 0.1 + {(0.1)}^{2}$

$\rm \: = \: 10000 + 20 + 0.01$

$\rm \: = \: 10020.01$

Hence,

$\green{\rm\implies \:\boxed{\rm{  \: {(100.1)}^{2} = 10020.01 \: \: }}}$

$\large\underline{\sf{Solution-c}}$

$\rm \: 51 \times 49$

$\rm \: = \: (50 + 1) \times (50 - 1)$

We know,

$\boxed{\rm{  \:(x + y)(x - y) = {x}^{2} - {y}^{2} \: \: }}$

So, here

$\rm \: x \: = \: 50$

$\rm \: y \: = \: 1$

So, on substituting the values in above identity, we get

$\rm \: = \: {(50)}^{2} - {(1)}^{2}$

$\rm \: = \: 2500 - 1$

$\rm \: = \: 2499$

Hence,

$\blue{\rm\implies \:\boxed{\rm{  \:51 \times 49 \: = \: 2499 \: \: }}}$

$\rule{190pt}{2pt}$

Additional Information :-

$\boxed{\rm{  \: {(x + y)}^{2} - 4xy = {(x - y)}^{2} \: }}$

$\boxed{\rm{  \: {(x - y)}^{2} + 4xy = {(x + y)}^{2} \: }}$

$\boxed{\rm{  \: {(x + y)}^{2} - {(x - y)}^{2} = 4xy \: \: }}$

$\boxed{\rm{  \: {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2}) \: \: }}$

Answer 2

$\underline{\huge \tt{ \red{A} \blue{n} {S} \green{w} \pink{E} \orange{r}}}$

$\\ \bf \: a) \: \: \: {99}^{2} \\ \\ \sf \: (100 - 1 {)}^{2} \\ \\ \bigstar \boxed{ \mathfrak{ {(a - b)}^{2} = {a}^{2} - 2ab + {b}^{2} }}$

Here ,

• a = 100
• b = 1

$\longmapsto \sf \: {100}^{2} - 2(100)( 1) + {1}^{2} \\ \longmapsto\sf \: 10000 - 200 + 1 \\ \longmapsto \sf \: \red{9801}$

__________

$\\ \bf b) \: \: \: 100. {1}^{2} \\ \\ \sf \: (100 + 0. {1)}^{2} \\ \\ \bigstar \boxed{ \mathfrak{(a { + b)}^{2} = {a}^{2} + 2ab + {b}^{2} }}$

Here ,

• a = 100
• b = 0.1

$\longmapsto \sf \: {100}^{2} + 2(100)(0 .1) + { 0.1}^{2} \\ \longmapsto \sf \: 10000 + 20 + 0.01 \\ \longmapsto \sf \: \red{10020.01}$

__________

$\\ \bf \: c) \: \: 49 \times 51 \\ \\ \sf \: (50 - 1)(50 + 1) \\ \\ \bigstar \boxed{ \mathfrak{(x - a)(x + a) = {x}^{2} - {a}^{2} }}$

Here ,

• x = 50
• a = 1

$\\ \longmapsto \sf \: {50}^{2} - {1}^{2} \\ \longmapsto \sf \: 2500 - 1 \\ \longmapsto \sf \red{ 2499}$