If $ab=100$ and $a+b=25$, find the value of $a^2+b^2$.

Given: $ab=100$ and $a+b=25$.

To do: To find the value of $a^2+b^2$.

Solution:

As given, $ab=100$ and $a+b=25$.

$\because ( a+b)^2=a^2+b^2+2ab$

$\Rightarrow a^2+b^2=( a+b)^2-2ab$

$\Rightarrow a^2+b^2=25^2+2\times100$

$\Rightarrow a^2+b^2=625-200$

$\Rightarrow a^2+b^2=525$

Therefore, $a^2+b^2=525$.