Question

Difference of two perfect square numbers is 23. If the difference of square root of these perfect square numbers is 1, find the sum of the square root of the perfect square numbers.

Answer 1

Let $a^2$ and $b^2$ are two perfect squares whose square roots are $a$ and $b,$ respectively.

The subtraction of the square root of the perfect square numbers is $1.$
$\Rightarrow a-b=1\Rightarrow a=b+1$

It is given that $a^2-b^2=23$
$\Rightarrow (b+1{)}^2-b^2=23$
$\Rightarrow (b+1)\times (b+1)-b^2=23$
$\Rightarrow b^2+b+b+1-b^2=23$
$\Rightarrow 2b+1-1=23-1$
$\Rightarrow 2b=22$
$\Rightarrow b=\frac{22}{2}$
$\Rightarrow b=11$
And, $a=b+1=11+1=12$

$\therefore$ The sum of the square roots are $a+b=12+11=23$