SOLUTION: Find two real numbers that have a sum of 8 and a product of 2.

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Question 153155: Find two real numbers that have a sum of 8 and a product of 2.
Answer by nerdybill(7384) About Me  (Show Source):
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Find two real numbers that have a sum of 8 and a product of 2.
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Let x = one of two real numbers
and y - second of two real numbers
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Since we have two unknowns, we'll need two equations.
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From:"that have a sum of 8" we get equation 1:
x + y = 8
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From:"product of 2" we get equation 2:
xy = 2
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Solving equation 1 for y:
x + y = 8
y = 8-x
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Plug the above into equation 2 and solve for x:
xy = 2
x(8-x) = 2
8x-x^2 = 2
0 = x^2 - 8x + 2
Since, we can't factor, we must use the quadratic equation.
x = {7.742, 0.258}
y = (0.258, 7.742)
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Answer: the two numbers are 7.742 and 0.258
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Details of quadratic solution follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-8x%2B2+=+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=%28-8%29%5E2-4%2A1%2A2=56.

Discriminant d=56 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--8%2B-sqrt%28+56+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-8%29%2Bsqrt%28+56+%29%29%2F2%5C1+=+7.74165738677394
x%5B2%5D+=+%28-%28-8%29-sqrt%28+56+%29%29%2F2%5C1+=+0.258342613226059

Quadratic expression 1x%5E2%2B-8x%2B2 can be factored:
1x%5E2%2B-8x%2B2+=+1%28x-7.74165738677394%29%2A%28x-0.258342613226059%29
Again, the answer is: 7.74165738677394, 0.258342613226059. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-8%2Ax%2B2+%29