SOLUTION: Find two numbers such that their product and sum equal 20.

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Question 373941: Find two numbers such that their product and sum equal 20.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
x*y = 20
x + y = 20
Subtracting x from each side of the second equation we get:
y = 20 - x
Substituting this expression for y into the first equation we get:
x*(20-x) = 20
Simplifying we get:
20x-+x%5E2+=+20
This is a quadratic equation so we want one side to be zero. Since it is also nice to have a positive squared term, I'm going to add x%5E2 and subtract 20x:
0+=+x%5E2+-+20x+%2B+20
This will not factor but we can use the Quadratic Formula:
x+=+%28-%28-20%29+%2B-+sqrt%28%28-20%29%5E2+-+4%281%29%2820%29%29%29%2F2%281%29
This simplifies as follows:
x+=+%28-%28-20%29+%2B-+sqrt%28400+-+4%281%29%2820%29%29%29%2F2%281%29
x+=+%28-%28-20%29+%2B-+sqrt%28400+-+80%29%29%2F2%281%29
x+=+%28-%28-20%29+%2B-+sqrt%28320%29%29%2F2%281%29
x+=+%2820+%2B-+sqrt%28320%29%29%2F2
x+=+%2820+%2B-+sqrt%2864%2A5%29%29%2F2
x+=+%2820+%2B-+sqrt%2864%29%2Asqrt%285%29%29%2F2
x+=+%2820+%2B-+8%2Asqrt%285%29%29%2F2
x+=+%282%2810+%2B-+4%2Asqrt%285%29%29%29%2F2
x+=+%28cross%282%29%2810+%2B-+4%2Asqrt%285%29%29%29%2Fcross%282%29
x+=+10+%2B-+4%2Asqrt%285%29
So x+=+10+%2B+4%2Asqrt%285%29 or x+=+10+-+4%2Asqrt%285%29. Each of these x's is just one of the two numbers. The other number, if you remember, is y which is 20-x. So
y+=+20+-%2810+%2B+4%2Asqrt%285%29%29 or y+=+20+-+%2810+-+4%2Asqrt%285%29%29.
Simplifying we get:
y+=+10+-+4%2Asqrt%285%29%29 or y+=+10+%2B+4%2Asqrt%285%29%29. So we have two pairs of numbers:
x+=+10+%2B+4%2Asqrt%285%29 and y+=+10+-+4%2Asqrt%285%29
or
x+=+10+-+4%2Asqrt%285%29 and y+=+10+%2B+4%2Asqrt%285%29
These are really the same pairs which, since addition and multiplication are Commutative, should be no surprise.