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 April 6th, 2016, 03:32 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, 03:38 PM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 2,815 Thanks: 1458 simultaneous equations ... $k+p=20 \implies k = 20-p$ $kp = 91 \implies (20-p)p = 91$ $(20-p)p = 91$ $0 = p^2 - 20p + 91$ factor the quadratic ... $0 = (p - 13)(p - 7)$ $p$ could be 13, which means $k=7$ or $p$ could be 7, which means $k=13$ Thanks from topsquark
 April 6th, 2016, 03:41 PM #3 Newbie   Joined: Apr 2016 From: Waterbury, CT Posts: 2 Thanks: 0 Thank you for your fast and clear answer !
April 6th, 2016, 03:41 PM   #4
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Quote:
 Originally Posted by Ruben715 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 !
$\displaystyle k + p = 20 \implies k = 20 - p$

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, 05:50 PM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,599 Thanks: 2587 Math Focus: Mainly analysis and algebra There's no need to calculate $k$ or $p$: $k^2+p^2=(k+p)^2-2kp=20^2-2\times 91=400-182=218$ Thanks from 123qwerty and aurel5

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