# The sum of two numbers is 25, the sum of their squares is 313. What are the numbers?

May 26, 2017

$13$ and $12$

#### Explanation:

If $x + y = 25$ then $y = 25 - x$.

Substitute:
${x}^{2} + {y}^{2} = 313$
${x}^{2} + {\left(25 - x\right)}^{2} = 313$
$2 {x}^{2} - 50 x + 312 = 0$
${x}^{2} - 25 x + 156 = 0$
$\left(x - 13\right) \left(x - 12\right) = 0$

$x = 13 , y = 12$
$x = 12 , y = 13$

May 26, 2017

$12$ and $13$.

#### Explanation:

Suppose the two numbers are $a$ and $b$

"The sum of two numbers is 25" gives us:

$a + b = 25$

"sum of their squares is 313" gives us:

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

From the first equation we have:

$b = 25 - a$

Substituting for $b$ into the second we get:

${a}^{2} + {\left(25 - a\right)}^{2} = 313$

$\therefore {a}^{2} + 625 - 50 a + {a}^{2} = 313$
$\therefore 2 {a}^{2} - 50 a + 312 = 0$
$\therefore {a}^{2} - 25 a + 156 = 0$
$\therefore \left(a - 12\right) \left(a - 13\right) = 0$
$\therefore a = 12 , 13$

We now use the second equation to find the value of $b$ corresponding to each solution.

$a = 12 \implies b = 25 - 12 = 13$
$a = 13 \implies b = 25 - 13 = 12$

So there is only one solution which is that the two numbers are $12$ and $13$.

May 26, 2017

The numbers are $12$ and $13$

#### Explanation:

Let te numbers be $x$ and $y$

therefore we have $x + y = 25$ ............(1)

which gives us ${x}^{2} + {y}^{2} + 2 x y = 625$ ............(2)

and we also have ${x}^{2} + {y}^{2} = 313$ ............(3)

Subtracting (3) from (2), we get $2 x y = 312$ ............(4)

and (4) from (3) we get

${x}^{2} + {y}^{2} - 2 x y = 1$ ............(5)

i.e. $x - y = 1$ ............(6)

Slolving for $x$ and $y$ from (1) and (6), we get

$x = 13$ and $y = 12$

Note:we can also have $x - y = - 1$, which gives $x = 12$ and $y = 13$