Question 1180203: the sum of the squares of two consecutive numbers is 61.What are those numbers Found 3 solutions by Boreal, mananth, ikleyn:Answer by Boreal(15235) (Show Source):
You can put this solution on YOUR website! x^2+(x+1)^2=61
x^2+x^2+2x+1=61
so 2x^2+2x-60=0
so x^2+x-30=0
(x+6)(x-5)=0
x=-6; and -6 and -5 have squares who add to 61.
x=5, and 5 and 6 do the same
Those are the two pairs of numbers.
You can put this solution on YOUR website! Let the consecutive numbers be x, x+1
x^2 +(x+1)^2 =61
x^2 +x^2 +2x +1 =61
2x^2 +2x -60 =0
divide by 2
x^2 +x -30=0
(x+6)(x-5) =0
x= -6 Or x=5
If one number is -6 other number is -5
If one number is 5 other number is 6
You can put this solution on YOUR website! .
the sum of the squares of two consecutive numbers is 61. What are those numbers
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Two other tutors provided a formal algebra solution.
Which is good to develop your technique.
But I will give here simple MENTAL solution - - - which is good to develop your mind.
The sum of the squares of two consecutive integer numbers is 61.
So, we can expect that each of the two squares is about 30 (about half of 61).
Such two squares of integers closest to 30 are 25 and 36 - so check 25 + 36 = 61 <<<---=== correct !
Thus the solution is the pair (5,6).
Recalling the signs, we detect another pair (-6,-5), too.
Solved MENTALLY.
Many other similar problems can be solved similarly.
It is a good technique to use it in competitions for fastest solution.
