SOLUTION: the sum of the squares of six consective integers is 1111.

Question 219375: the sum of the squares of six consective integers is 1111.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Consecutive integers are 1 apart from each other. So if we call the smallest of our six consecutive integers "x", then the next one would be "x+1". And if the second one is "x+1" then the third one is "x+1+1" = "x+2". Continuing with this our six consecutive integers are x, x+1, x+2, x+3, x+4 and x+5. The squares of these integers are , , , , , .

Now we are ready to translate "the sum of the squares of six consecutive integers is 1111" into an equation:

To solve this we will start by simplifying the left side. (Use FOIL or the pattern ):

Adding like terms:

Since this is a quadratic equation we will now get one side equal to zero by subtracting 1111 from each side:

Now we can either use factoring or the quadratic formula to solve. Factoring, when it can be done, is often easier so we will start by trying to factor this. Whenever you factor, start with the Greatest Common Factor (GCF). The GCF here is 6. Factoring out 6 we get:

Next we want to factor . This is a trinomial (3 term expression.) with a leading coefficient of 1. These trinomials will factor if you can answer the question: "What factors of the constant term (at the end) add up to the coefficient of the middle term?" In this problem the question is "What factors of -176 add up to 5? Since product of the factors is negative (-176) there must be one positive factor and one negative factor. And since the positive and negative factors add up to a positive (5), the positive factor must be the "larger" factor. With this information and some trial and error (or a good knowledge of multiplication facts) we should be able to find that the factors are 16 and -11. (Multiply them and you get -176. Add them and you get 5.) So factors into . So now we have:

The only way for this product to be zero is if one of the factors is zero. (Zero Product Property). 6 cannot ever be zero but (x+16) and (x-11) could be with the right values of x. So we solve:
or
which gives us:
or
Remember that "x" stands for the smallest of the six consecutive integers. So our six consecutive integers are: -16, -15, -14, -13, -12, -11 or 11, 12, 13, 14, 15, 16.
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