Proving a quadratic equation has no integral roots Show

Pablo Dennis

Pablo Dennis

Answered question

2022-03-28

Proving a quadratic equation has no integral roots
Show that the quadratic equation x27x14(q2+1)=0 where q is an integer, has no integral real roots.

Answer & Explanation

Abdullah Avery

Abdullah Avery

Beginner2022-03-29Added 19 answers

To prove that no perfect square k2=7n+6, we work with k=7m±l;l0,1,2,3
(7m+0)2=49m2=7(7m2)+0
(7m±1)2=49m2±14m+1=7(7m2±2m)+1
(7m±2)2=49m2±28m+4=7(7m2±4m)+4
(7m±3)2=49m2±42m+9=7(7m2±6m+1)+2
None of these are of the form 7n+6, thusly it is not possible

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