Square is the way

Number Theory Level 4

Find all the integer values for n n such that n 2 + 20 n + 11 n^2+20n+11 is a perfect square. Give the answer as n 1 2 + n 2 2 + … + n k 2 n_1^2+n_2^2+\ldots+n_k^2 .


The answer is 4250.

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Drop TheProblem by Drop TheProblem , 24, Italy

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2 solutions

Drop TheProblem
Aug 10, 2015

n 2 + 20 n + 11 = k 2 n^2+20n+11=k^2

n 2 + 20 n + 11 + 100 − 100 = k 2 n^2+20n+11+100-100=k^2

n 2 + 20 n + 100 − 89 = k 2 n^2+20n+100-89=k^2

( n + 10 ) 2 − k 2 = 89 (n+10)^2-k^2=89

( n + 10 − k ) ( n + 10 + k ) = 89 (n+10-k)(n+10+k)=89

1) { n + 10 − k = 89 n + 10 + k = 1 \begin{cases} n+10-k=89\\n+10+k=1 \end{cases}

2) { n + 10 + k = 89 n + 10 − k = 1 \begin{cases} n+10+k=89\\n+10-k=1 \end{cases}

3) { n + 10 − k = − 89 n + 10 + k = − 1 \begin{cases} n+10-k=-89\\n+10+k=-1 \end{cases}

4) { n + 10 + k = − 89 n + 10 − k = − 1 \begin{cases} n+10+k=-89\\n+10-k=-1 \end{cases}

For system 1) and 2) → n = 35 , k 2 = 1936 \rightarrow n=35, k^2=1936

For system 3) and 4) → n = − 55 , k 2 = 1936 \rightarrow n=-55, k^2=1936

In conclusion 3 5 2 + ( − 55 ) 2 = 4250 35^2+(-55)^2=\boxed{4250}

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why did you add and subtract 100 in the second step??????????????

bhumika sharma - 5 years, 9 months ago

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he is completing the square

sumit cooooool - 5 years, 9 months ago

If he does not subtract 100 then we will have 100 as constant term not 11

Satyajit Ghosh - 5 years, 7 months ago

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She didn't mean to ask that. She meant that why only 100? And that's because only 100 will complete the square as n^2+20n+100=(n+10)^2

Kushagra Sahni - 5 years, 2 months ago

S t a n d a r d m e t h o d . f ( x ) = x 2 + 2 a x + b , c o n s t a n t s i n t e g e r s , a , b g i v e n . F i n d v a l u e s o f x t h a t m a k e s f ( x ) = k 2 a p e r f e c t s q u a r e . B e i n g a q u a d r a t i c t h e r e w i l l b e t w o s o l u t i o n s . f ( x ) = ( x + a ) 2 − k 2 = ( a 2 − b ) . F a c t o r ( a 2 − b ) i n t o t w o f a c t o r s . M a n d N ( M o r N c a n e v e n b e = ± 1. ) C o m p a r e t h e s e t w o f a c t o r s w i t h t h e s e t w o ( x + a + k ) ( x + a − k ) a n d g e t t w o e q u a t i o n s ( x + a + k ) = M a n d ( x + a − k ) = N a n d v a l u e o f x 1 f o u n d a n d t h e n o f k 1 ∴ k 1 2 . a n d , ( x + a + k ) = − M a n d ( x + a − k ) = − N a n d t h e s e c o n d v a l u e o f x 2 f o u n d a n d t h e n o f k 2 ∴ k 2 2 . I n t h i s p r o b l e m a = 10 , a n d b = 11. M = 89 , N = 1. S o l v i n g , x 1 = 35 , x 2 = − 55. Standard~ method.~~f(x)=x^2+2ax +b, ~constants~ integers, ~a,~b~given.~~Find~values ~of~x~that~makes~f(x)=k^2~ a ~perfect~square.\\ Being~a~quadratic~there~will~be~two~solutions.\\ f(x)~=~(x+a)^2-k^2=(a^2-b).~~\\ Factor~(a^2-b)~into~ two~ factors.~M~and~N~~~(M~or~N~~can~even~be~=\pm 1.)\\ Compare~these~two~factors~with~these~two~(x+a+k)(x+a-k)~and~get~~two~equations~~\\ (x+a+k)=M~~and~~(x+a-k)=N~and~value~of~x_1~found~and~ then~of~k_1~\therefore~k_1^2.~~and,\\ (x+a+k)= - M~~and~~(x+a-k)= - N~and~the~second~value~of~x_2~found~and~ then~of~k_2~\therefore~k_2^2.\\ In~this~problem~~a=10,~and~b=11.~~M=89,~~N=1.~~Solving,~~x_1=35,~~x_2= - 55.

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