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April 6th, 2016, 02:32 PM  #1 
Newbie Joined: Apr 2016 From: Waterbury, CT Posts: 2 Thanks: 0  Can you explain to me in details ?
QUESTION : Two integers k and p check the relationship k + p = 20 and k * p = 91. What is the value of k² + p² ? ANSWER : k and p have the values 13 and 7, hence k² + p² = 13² + 7² = 218. I still do not understand why k = 13 and p = 7. Could you give me a hint or explain to me in details please ? Thank you in advance for your answer ! 
April 6th, 2016, 02:38 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,947 Thanks: 1555 
simultaneous equations ... $k+p=20 \implies k = 20p$ $kp = 91 \implies (20p)p = 91$ $(20p)p = 91$ $0 = p^2  20p + 91$ factor the quadratic ... $0 = (p  13)(p  7)$ $p$ could be 13, which means $k=7$ or $p$ could be 7, which means $k=13$ 
April 6th, 2016, 02:41 PM  #3 
Newbie Joined: Apr 2016 From: Waterbury, CT Posts: 2 Thanks: 0 
Thank you for your fast and clear answer !

April 6th, 2016, 02:41 PM  #4  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,200 Thanks: 898 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
So $\displaystyle kp = 91 \implies (20  p)p = 20p  p^2 = 91$ Solve for p. Then k = 20  p. Note: We get twin solutions (p, k) = (13, 7) and (p, k) = (7, 13). Obviously these are equivalent. Dan  
April 6th, 2016, 04:50 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,663 Thanks: 2643 Math Focus: Mainly analysis and algebra 
There's no need to calculate $k$ or $ p$: $k^2+p^2=(k+p)^22kp=20^22\times 91=400182=218$ 

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