Question

two natural numbers differ by two units and the sum of their squares is 580

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{Difference oftwo natural numbers is 2 and sum their squares is 580}$

$\underline{\textsf{To find:}}$

$\textsf{The natural numbers}$

$\underline{\textsf{Solution:}}$

$\textsf{Let the two natrual numbers be x and y with x >y }$

$\textsf{As per given data,}$

$\mathsf{x-y=2}$

$\mathsf{x=y+2}$.......(1)

$\mathsf{x^2+y^2=580}$......(2)

$\textsf{Using (1) in (2), we get}$

$\mathsf{(y+2)^2+y^2=580}$

$\mathsf{y^2+4+4y+y^2=580}$

$\mathsf{2y^2+4y-576=0}$

$\implies\mathsf{y^2+2y-288=0}$

$\textsf{Factorise, we get}$

$\mathsf{(y+18)(y-16)}$

$\implies\mathsf{y=-18,16}$

$\textsf{Since y is natural number, it cannot be negative}$

$\implies\mathsf{y=16}$

$\textsf{When y=16, x=16+2=18}$

$\therefore\textsf{The required two natural numbers are 16 and 18}$

Find more:

The difference between two natural numbers is 73 . If the greatest number is divided by the smaller number, the quotient is 4 and the remainder is 4.Find the numbers.

https://brainly.in/question/7027379

Answer 2

${\huge{\underline{\underline{\red{Given Data :- }}}}}$

Difference of Two Natural Numbers is 2 and The sum of Squares of those numbers is 580 .

${\huge{\underline{\underline{\blue{to \: find :- }}}}}$

Those Natural Numbers .

${\huge{\underline{\underline{\green{solution:- }}}}}$

Let the numbers be denoted as “ x ” & “ y ” with x > y .

According to Given Data ,

$x - y = 2$

$x = 2 + y \: \: ......( \: eq \: 1 \: )$

${x}^{2} + {y}^{2} = 580 \: \: ....... \: ( \: eq \: 2 \: )$

Now substitute eq1 in eq2 , we get

$({2 + y})^{2} + {y}^{2} = 580$

${y}^{2} + 4 + 4y + {y}^{2} = 580$

${2y}^{2} + 4y - 576 = 0$

${y}^{2} + 2y - 288 = 0$

${\huge{\orange{factorise \: we \: get }}}$

$(y + 18) \: \: (y - 16) = 0$

$⟹ \: y = - 18 \: \: or \: \: 16$

∴ Hence , y is a Natural Number it can't be in Negative so y = 16

$now \: \: substitute \: y = 16 \: in \: eq1 \: \: we \: get$

$x = 2 + 16$

$x = 18$

Hence , The Natural Numbers are 16 & 18 .

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