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?

How do I solve it?

Assign letters to the unknown integers.

Let x be the first one, and y the second one.

From the first piece of info;



From the second piece of info;



It's a simultaneous equation. Find their values, then you'll be able to write down the sum of the two integers. :)

But the problem is that I haven't studied quadratic equations yet. How then, I know what is xy=84?

84/x=y?

Oh, I thought you understood these... sorry.. :o

Ok, here we go! :)





Yes, start by making one of them the subject of formula.



Then, replace it in the first equation. You now know the 'equivalent' of y, replace all y by that equivalent:






Multiply everything by x^2 to remove fractions;



Now rearrange to the same side:





Therefore, x = 7 or 12.

And so is y.

And the sum of the integers is (12 + 7)= 19!

:)

Therefore, x = 7 or 12.

And so is y.

And the sum of the integers is (12 + 7)= 19!


Umm, please go back and explain how you got from x^4 to x^2 and how you got 49 and 144 please. Where did 7056 goo?

Rewrite it as

Then, factor.

What two numbers when multiplied equal 7056 and when added equal -193?

How about -144 and -49

(-144)(-49)=7056

-144+(-49)=-193









Now, because it was an x^4 in the beginning, put an x^2 back in place of the x



Note these are the difference of two squares:



But

So, we have

To add to galactus explanation, I'll put it like this.

You let for example

What will you have?



If a = x^2, then a^2 = x^4. So;



Then factorise normally, to have:



Then, replace back to x^2, you have:



And, solving, you have:







But since the integers are only positive, you consider only 7.

The same thing goes for the other integer. :)

THanks guys =)

You're welcomed suvivorboi! Care to take a look at my thread? There are lots of challenging questions! :)

https://www.askmehelpdesk.com/mathem...ns-384108.html