SOLUTION: Please help. I need to find three consectutive integers such that the sum of their squares is 77. Thanks!!

Question 87338: Please help. I need to find three consectutive integers such that the sum of their squares is 77. Thanks!!
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let x=1st #, y=2nd #, z=3rd #

This is the translation of "the sum of their (each number) squares is 77"

Since each number is consecutive, we're really adding 1 each time. So

and

Plug in and

Foil

Combine like terms

Subtract 77 from both sides

Combine like terms

Now lets use the quadratic formula to solve for x

Starting with the general quadratic

the general form of the quadratic equation is:

So lets solve

Plug in a=3, b=6, and c=-72

Square 6 to get 36

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root

Multiply 2 and 3 to get 6

So now the expression breaks down into two parts

or

Lets look at the first part:

Add the terms in the numerator
Divide

So one answer is

Now lets look at the second part:

Subtract the terms in the numerator
Divide

So another answer is

So our solutions are:
or

Notice when we graph we get:

and we can see that the roots are and . This verifies our answer

So our first number could be 4 or -6

Lets use to find the 2nd and 3rd number
and
and
So we have 3 numbers: 4,5,6

Check:

works

So the 3 numbers 4,5,6 work

Now lets use to find the 2nd and 3rd number
and
and
So we have 3 numbers: -6,-5,-4

Check:

works

So the 3 numbers -6,-5,-4 work

Answer:
So if you allow negative numbers, you get 2 possible answers
4,5,6
or
-6,-5,-4

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