SOLUTION: Find the exact solution(s) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth. More missing numbers. Find two real n

Algebra ->  Real-numbers -> SOLUTION: Find the exact solution(s) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth. More missing numbers. Find two real n      Log On


   

Question 198350: Find the exact solution(s) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth.
More missing numbers. Find two real numbers that have a sum of 8 and a product of 2.
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find two real numbers that have a sum of 8 and a product of 2.
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x*(8-x) = 2
x%5E2+-+8x+%2B+2+=+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-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+=+%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

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x = 4+sqrt(14) and 4-sqrt(14)
x = 7.74166, 0.25834
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x = 7.7, 0.3