SOLUTION: find two numbers whose sum is 55 and whose product is 684

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Question 162375: find two numbers whose sum is 55 and whose product is 684

Answer by Alan3354(30956) About Me  (Show Source):
You can put this solution on YOUR website!
find two numbers whose sum is 55 and whose product is 684
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Call the numbers x and y.
x+y = 55
xy = 684
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From the 1st eqn, y = 55-x
Sub that into the 2nd eqn
x*(55-x) = 684
55x - x^2 = 684
x^2 - 55x + 684 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-55x%2B684+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-55%29%5E2-4%2A1%2A684=289.

Discriminant d=289 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--55%2B-sqrt%28+289+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-55%29%2Bsqrt%28+289+%29%29%2F2%5C1+=+36
x%5B2%5D+=+%28-%28-55%29-sqrt%28+289+%29%29%2F2%5C1+=+19

Quadratic expression 1x%5E2%2B-55x%2B684 can be factored:
1x%5E2%2B-55x%2B684+=+%28x-36%29%2A%28x-19%29
Again, the answer is: 36, 19. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-55%2Ax%2B684+%29

The graph isn't much help, but the numbers are given:
19 and 36