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April 6th, 2016, 03:32 PM   #1
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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 !
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April 6th, 2016, 03:38 PM   #2
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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 $k=13$
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April 6th, 2016, 03:41 PM   #3
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Thank you for your fast and clear answer !
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April 6th, 2016, 03:41 PM   #4
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Quote:
Originally Posted by Ruben715 View Post
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
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April 6th, 2016, 05:50 PM   #5
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There's no need to calculate $k$ or $ p$:
$k^2+p^2=(k+p)^2-2kp=20^2-2\times 91=400-182=218$
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