Question

If $a$ and $b$ are two consecutive odd positive integers and sum of their squares is $290$, then value of $a+b$ is

Answer 1

Let the smaller of two be $a$, so
$a+2=b$
Given condition
$a^2+b^2=290\Rightarrow a^2+(a+2{)}^2=290\Rightarrow 2a^2+4a-286=0\Rightarrow a^2+2a-143=0\Rightarrow a^2+13a-11a-143=0\Rightarrow a(a+13)-11(a+13)=0\Rightarrow (a-11\left)\right(a+13)=0\Rightarrow a=-13,11\therefore a=11(\because a\text{is positive integer})\Rightarrow b=11+2=13\therefore a+b=11+13=24$