The sum of two numbers is 25 and the sum of their squares is 313. How do you find the numbers?

Mar 2, 2018

$12$ and $13$

Explanation:

let,the two numbers be $a$ and $b$,

So, $a + b = 25$

and, ${a}^{2} + {b}^{2} = 313$

Now, ${a}^{2} + {b}^{2} = {\left(a + b\right)}^{2} - 2 a b$

so, $313 = 625 - 2 a b$

so, $a b = 156$

Now, ${\left(a - b\right)}^{2} = {\left(a + b\right)}^{2} - 4 a b$

or, ${\left(a - b\right)}^{2} = 625 - 624 = 1$

So, $\left(a - b\right) {=}_{-}^{+} 1$

So, we have, $a + b = 25$ and, $a - b {=}_{-}^{+} 1$

Solving both we get, $a = 13. b = 12$ and $a = 12 , b = 13$

So,the numbers are 12&13