Question

find two consecutive odd integers,sum of whose square is 970​

Answer 1

Answer:

21 , 23

Step-by-step explanation:

To find---> Two consecutive odd integers , sum of whose square is 970.

Solution---> We know that, difference between two consecutive odd integers 2 .

So, let two consecutive odd integers be x and

( x + 2 ).

ATQ, Sum of two consecutive odd integers is 970.

So, x² + ( x + 2 )² = 970

We know that,

( a + b )² = a² + b² + 2ab , applying it here, wr get,

=> x² + x² + 4 + 4x = 970

=> 2 x² + 4x + 4 - 970 = 0

=> 2 x² + 4 x - 966 = 0

Dividing by whole equation by 2 ,

=> x² + 2 x - 483 = 0

Now we , factorize it by splitting the middle term method.

=> x² + ( 23 - 21 ) x - 483 = 0

=> x² + 23 x - 21 x - 483 = 0

=> x ( x + 23 ) - 21 ( x + 23 ) = 0

=> ( x + 23 ) ( x - 21 ) = 0

If ( x + 23 ) = 0

x = - 23 ( impossible )

If ( x - 21 ) = 0

=> x = 21

First odd number = x

= 21

Second odd number = x + 2

= 21 + 2

= 23

Answer 2

$\bf{\Huge{\boxed{\bf{\green{ANSWER\::}}}}}$

Given :

Sum of whose square is 970.

To Find :

Two consecutive odd integers.

$\bf{\Large{\underline{\sf{\blue{Explanation\::}}}}}$

Let the two consecutive odd integers is R & R+2.

Therefore,

$\longmapsto\tt{R^{2} +(R+2)^{2} =970}$

$\longmapsto\tt{R^{2} +(R)^{2} +(2)^{2} +2*R*2=970}$

$\longmapsto\tt{R^{2} +R^{2}+4+4R=970}$

$\longmapsto\tt{2R^{2} +4+4R=970}$

$\longmapsto\tt{2R^{2} +4R-966=0}$

$\longmapsto\tt{2(R^{2} +2R-483)=0}$

$\longmapsto\tt{R^{2} +4R-483=\cancel{\frac{0}{2} }}$

$\longmapsto\tt{R^{2} +2R-483=0}$

Using Sridharacharya Formula:

$\leadsto{\sf{x\:=\:\frac{-b ± \sqrt{b^{2}-4ac} }{2a}} }$

So,

$\leadsto\tt{x\:=\:\frac{-2 ± \sqrt{(2)^{2}-4(1) (-483)} }{2} }$

$\leadsto\tt{x\:=\:\frac{-2 ± \sqrt{4+1932 } }{2} }$

$\leadsto\tt{x\:=\:\frac{-2 ± \sqrt{1936} }{2} }$

$\leadsto\tt{x\:=\:\frac{-2 ± 44}{2} }$

So,

$\leadsto\tt{x\:=\:\frac{-2+44}{2} \:\:\:\:\:\:\:or\:\:\:\:\:\:x\:=\:\frac{-2-44}{2} }$

$\leadsto\tt{x\:=\:\cancel{\frac{42}{2} }\:\:\:\:\:\:or\:\:\:\:\:\:x=\cancel{\frac{-46}{2} }}$

$\leadsto\tt{\red{x\:=\:21\:\:\:\:\:\:\:\:\:or\:\:\:\:\:\:\:\:\:x=-23}}$

We know that negative value is not acceptable.

Thus,

First consecutive number is 21.

Second consecutive number is 21+2 = 23.