SOLUTION: Find two numbers (exactly) whose product is 24 and whose sum is 21

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Question 524054: Find two numbers (exactly) whose product is 24 and whose sum is 21
Answer by Alan3354(69443) About Me  (Show Source):
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Find two numbers (exactly) whose product is 24 and whose sum is 21
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x + y = 21
x*y = 24
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y = 21-x
x*(21-x) = 24
12x-x%5E2+=+24
x%5E2+-+12x+%2B+24+=+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-12x%2B24+=+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=%28-12%29%5E2-4%2A1%2A24=48.

Discriminant d=48 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--12%2B-sqrt%28+48+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-12%29%2Bsqrt%28+48+%29%29%2F2%5C1+=+9.46410161513775
x%5B2%5D+=+%28-%28-12%29-sqrt%28+48+%29%29%2F2%5C1+=+2.53589838486225

Quadratic expression 1x%5E2%2B-12x%2B24 can be factored:
1x%5E2%2B-12x%2B24+=+%28x-9.46410161513775%29%2A%28x-2.53589838486225%29
Again, the answer is: 9.46410161513775, 2.53589838486225. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-12%2Ax%2B24+%29

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