Medal & Fan.
The sum of the squares of 3 consecutive positive integers is 116. What are the numbers?
Which of the following equations is used in the process of solving this problem?
3n^2 + 5 = 116
3n^2 + 3n + 3 = 116
3n^2 + 6n + 5 = 116

- anonymous

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- schrodinger

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- anonymous

@hybrik

- freckles

an and example of 3 consecutive positive integers is:
1,2,3
or
3,4,5
...
or
(n-1),n,(n+1)
we don't know what they are so let's go with the (n-1),n,(n+1) being the 3 consecutive positive integers
so you have the sum of the squares of them is 116
that is (n-1)^2+n^2+(n+1)^2=116
play with the left hand side

- hybrik

Freckles can do this one, I have to do something at the moment, be right back.

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## More answers

- anonymous

2n+2+n+2n+2=116?

- anonymous

probly horribly wrong

- freckles

(n-1)^2
=
(n-1)(n-1)
do you know how to expand this?

- anonymous

foil maybe?

- freckles

yeah
or it is just really distributing
like n(n-1)-1(n-1)

- anonymous

n^ 2- 2n + 1

- freckles

right
and we can also expand (n+1)^2 in a similar way

- freckles

which should be n^2+2n+1 for (n+1)^2
so you have this now:
\[(n-1)^2+n^2+(n+1)^2 =116 \\ (n^2-2n+1)+n^2+(n^2+2n+1)=116 \]

- freckles

combine like terms on the left hand side

- anonymous

\[-2n ^{2}+1+n^2 +2n^2+1=116\]

- anonymous

not done yet

- freckles

don't go any further with what you have just wrote

- freckles

in the equation I wrote how many n^2 's do you see?

- anonymous

3

- freckles

n^2+n^2+n^2 is 3n^2

- freckles

now let's look at the n's
you have -2n+2n which equals ?

- anonymous

n

- freckles

if someone gave you 3 apples and your three all 3 apples away how many apples do you have left?

- anonymous

0

- freckles

if someone gave you 3 apples and your threw all 3 apples away how many apples do you have left?*
yes 0

- freckles

so -2n+2n is ?

- anonymous

0

- freckles

0 also
replace 2n with 5
you have -5+5 and we know that is 0
you can also write this
-2n+2n
both terms have a common factor n
factor the n out
n(-2+2)
but you should know -2+2=0
so you have
n(0)
but 0 times anything will result in 0
so n(0)=0
so -2n+2n=0

- freckles

the last thing to add are the constant terms (the terms with out any variable)

- freckles

can you show me what you think you have?

- anonymous

the whole thing?

- freckles

sure why not
you probably want me to check the whole thing right?

- anonymous

3n^2+2=0?

- freckles

yep but that doesn't match any of your answers
I guess they chose their 3 consecutive numbers differently
maybe they chose n,n+1,n+2 which also works
\[n^2+(n+1)^2+(n+2)^2=116 \\ n^2+n^2+2n+1+n^2+4n+4=116 \\ 3n^2+6n+5=116 \]

- freckles

you can choose the three consecutive integers in many ways

- freckles

that is solving either one of the equations will result in the same 3 consecutive integers

- anonymous

thanks again

- freckles

though I like my equation better because it seems simpler to solve :p

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