Question

Find the two consecutive odd natural numbers, the sum of whose squares is $130$.

Answer 1

let the two consecutive odd natural number be $x$ and $x+2$

given $x^2+(x+2{)}^2$ =$x^2+x^2+4x+4=130$

$x^2+2x+2=65$

$x^2+2x-63=0$

$x^2+9x-7x-63=0$

$x(x+9)-7(x+9)=0$

$(x+9)(x-7)=0$

$x=7$ or $x=-9$

if $x=7,x+2=9$

if $x=-9,x+2=-7$

since $x$ the number should be natural $x\ne -9$

$\therefore$ $7$ and $9$ are two consecutive natural number