document.write( "Question 238720: What is the largest possible product of a set of positive integers whose sum is 20? \n" ); document.write( "
Algebra.Com's Answer #175404 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
I'm assuming the problem is about 2 positive integers and that you are not looking for a solution that uses Calculus. If I am wrong about either assumption then stop reading and be more specific when you repost your question.

\n" ); document.write( "When you do word problems, it is often to your advantage to use as few variables as possible. So if we say
\n" ); document.write( "Let x = one of the positive integers
\n" ); document.write( "we could use an entirely separate variable for the other positive integer. Or we could take advantage of the fact that the two integers add up to 20 and say
\n" ); document.write( "then (20-x) = the other integer
\n" ); document.write( "(If if is not clear to you why (20-x) works, Just add x and (20-x) and see what you get.) Using the x and (20-x), the product becomes:
\n" ); document.write( "x(20-x)
\n" ); document.write( "If we multiply this out we get:
\n" ); document.write( "\"20x+-+x%5E2\"
\n" ); document.write( "or
\n" ); document.write( "\"-x%5E2+%2B+20x\"
\n" ); document.write( "So the question is now, what is the largest value this can be? If we call this y then
\n" ); document.write( "\"y+=+-x%5E2+%2B+20x\"
\n" ); document.write( "and we have the equation of a parabola. Because of the negative coefficient in front of \"x%5E2\" this parabola opens downward. If you can picture such a parabola, you will realize that the highest point (which would be the highest product since that is what y represents) would be the vertex of this parabola. So if we can find the vertex we can find the value for x that makes the product the largest it can possibly be.

\n" ); document.write( "The vertex of a parabola can be found in different ways. One way is to use the fact that for the general parabola, \"y+=+ax%5E2+%2B+bx+%2B+c\", the vertex will be where x = -b/2a. For our equation, \"y+=+-x%5E2+%2B+20x\", b = 10 and a = (-1) so the vertex will be when x = -(20)/(2(-1)) = 10. And this makes the other positive integer, 20-x, also 10.

\n" ); document.write( "P.S. Another way to find the vertex is to transform the equation into the form:
\n" ); document.write( "\"y+-+k+=+4p%28x+-+h%29%5E2\"
\n" ); document.write( "In a parabola in this form, the vertex is (h, k). To transform \"y+=+-x%5E2+%2B+20x\", we start by factoring out -1:
\n" ); document.write( "\"y+=+-1%28x%5E2+-+20x%29\"
\n" ); document.write( "Next we complete the square in the parentheses. Since half of 20 is 10 and 10 squared is 100, we want the expression in the parentheses to be \"x%5E2+-20x+%2B+100\". In order to add the 100 inside the parentheses on the right side, we need to realize that, because of the -1 outside the parentheses, that we are actually adding -100 to the right side when we put a 100 inside the parentheses. And if we add -100 to the right side, we need to add -100 to the left side, too. This gives us:
\n" ); document.write( "\"y+%2B+%28-100%29+=+-1%28x%5E2+-20x+%2B+100%29\"
\n" ); document.write( "By completing the square we can now rewrite the expression in the parentheses as the perfect square we've created:
\n" ); document.write( "\"y+%2B+%28-100%29+=+-1%28x+-+10%29%5E2\"
\n" ); document.write( "Rewriting the left side as a subtraction we have:
\n" ); document.write( "\"y+-100+=+-1%28x+-+10%29%5E2\"
\n" ); document.write( "And we have the proper form. We can see that the vertex is (10, 100). And so, like our earlier solution, x = 10 and (20-x) = 10 are the two positive integers that add up to 20 and provide the largest possible product (100). \n" ); document.write( "
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