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

3 Answers
May 26, 2017

13 and 12

Explanation:

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

Substitute:
x^2+y^2=313
x^2+(25-x)^2=313
2x^2-50x+312=0
x^2-25x+156=0
(x-13)(x-12)=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+(25-a)^2=313

:. a^2+625-50a+a^2=313
:. 2a^2-50a+312=0
:. a^2-25a+156=0
:. (a-12)(a-13)=0
:. a=12,13

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

a=12 => b=25 - 12 = 13
a=13 => 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+2xy=625 ............(2)

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

Subtracting (3) from (2), we get 2xy=312 ............(4)

and (4) from (3) we get

x^2+y^2-2xy=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