SOLUTION: the sum of the squares of two consecutive positive even integers is 340. find the integers.

Question 110012This question is from textbook algebra(structure and method)
: the sum of the squares of two consecutive positive even integers is 340. find the integers. This question is from textbook algebra(structure and method)

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Begin by letting x equal one of the positive even integers. Then the next consecutive
positive even integer is x + 2 (because even numbers are 2 units away from the preceding
even integer.)
.
Square the two integers to get and . Then add these
two squares and get:
.

.
You can combine the two terms to simplify this to:
.

.
and you are told that this sum equals 340. In equation form this is:
.

.
You can simplify this a little by dividing both sides (each term) by 2 because 2 is a factor
of all the terms in this equation. Dividing by 2 results in:
.

.
Get this into standard quadratic form by subtracting 170 from both sides of the equation so
that the right side is zero. This subtraction leads to:
.

.
The left side of this equation can be factored. This factoring leads to:
.

.
This equation will be true if either of the factors is zero, because a multiplication
by zero on the left side results in the entire left side becoming zero and therefore
equal to the right side. So find out what values of x cause a factor to equal zero by
setting each factor equal to zero:
.

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Solve by subtracting 14 from both sides to get:
.

.
So one of the integers that could satisfy this equation is -14, but we can eliminate
this because the problem requires that the consecutive even integers be positive.
.
Setting the other factor equal to zero:
.

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Solve for x by adding 12 to both sides to get:
.

.
This answer tells us that the first positive even integer is 12, and therefore, then next
consecutive even integer is 14.
.
Check these answers. Does ? Yes it does because and .
So our check is and this means that our answer works.
.
The answer to this problem is that two consecutive positive even integers that work are 12 and 14
because when their squares are summed, the total is 340.
.
Hope this helps you to understand the problem.
.
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