Question 1111874
.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Let me show you the &nbsp;<U>RIGHT WAY</U>&nbsp; solving this problem.


<pre>
We are given

x + y = 1.


Cube both sides. You will get

1 = {{{x^3 + 3x^2y + 3xy^2 + y^3}}} = {{{(x^3 + y^3)}}} + {{{3xy(x+y)}}}.


Now replace {{{(x^3 + y^3)}}} by 19 ( ! given !)  and replace (x+y) by 1 in the right side.  You will get

1 = 19 + 3xy,    which implies  3xy = 1 - 19 = -18.


Hence, xy = {{{(-18)/3}}} = -6.


Now the last step is 

{{{x^2 + y^2}}} = {{{(x+y)^2 - 2xy}}} = 1 - 2xy = 1 - 2*(-6) = 1 + 12 = 13.
</pre>

<U>Answer</U>.  &nbsp;&nbsp;x^2 + y^2 = 13.


-------------------


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is how this problem MUST be solved, &nbsp;and it is how &nbsp;<U>IT &nbsp;IS &nbsp;EXPECTED</U> &nbsp;you will solve it &nbsp;&nbsp;(<U>if you know the way</U> !)


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Any other way is the way to &nbsp;<U>N O W H E R E</U>.


--------------


If you want to see other similar solved problems, look into the lesson


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-expressions-involving-x%2Binv%28x%29-x2%2Binv%28x2%29-and-x%5E3%2Binv%28x%5E3%29.lesson>HOW TO evaluate expressions involving &nbsp;{{{(x + 1/x)}}}, &nbsp;{{{(x^2+1/x^2)}}}, &nbsp;{{{(x^3 + 1/x^3)}}}  &nbsp;and &nbsp;{{{(x^5+1/x^5)}}}</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic "<U>Evaluation, substitution</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.


==============


Again: &nbsp;&nbsp;This problem is to check your competency: &nbsp;whether you know the way . . . 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is very good for an interview . . . (on a mathematical position . . . )