Question

The sum of the squares of two positive integers is 193. the product of the two integers is 84. what is the sum of the two integers? your answer submitgive up

Answer 1

The numbers are 7 and 12. 

This is because you can use the system of equations below. 

x^2 + y^2 = 193
xy = 84

Solve using substitution. 

Answer 2

Answer:

Let the integers be x and y

The sum of the squares of two positive integers is 193.

$x^{2} +y^{2} =193$

The product of the two integers is 84.

$xy=84$ or

$2xy=168$   ......(2)

Adding both equations (1) and (2)

$x^{2} +y^{2}+2xy =361$

So, we get $(x+y)^{2} =361$

$\sqrt{(x+y)^{2}} =\sqrt{361}$

$x+y=19$

So, the sum of the two positive integers is 19.

Now we get; $x=19-y$

Substituting in xy=84;

$(19-y)y=84$

=> $19y-y^{2}=84$

Equating to zero, we get;

=> $y^{2} -19y+84=0$

=> $y^{2} -12y-7y+84=0$

=> $y(y-12)-7(y-12)=0$

=> $(y-7)(y-12)=0$

Both are positive roots and the answer.

Hence, the two numbers are 7 and 12.

We can check :

$(7)^{2}+(12)^{2}=193$

$49+144=193$

$193=193$

And

$7\times12=84$

And sum =$7+12=19$