SOLUTION: Where do the two graphs {{{f(x)=x^2+2}}} and {{{g(x)=x-8}}} intersect?

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Question 438459: Where do the two graphs f%28x%29=x%5E2%2B2 and g%28x%29=x-8 intersect?
Found 3 solutions by Gogonati, Alan3354, josmiceli:
Answer by Gogonati(855) About Me  (Show Source):
You can put this solution on YOUR website!
To find the intersection points we solve the equation: x%5E2%2B2=x-8 or
x%5E2-x%2B10=0, since D<0, this equation has not real root, therefore these two
graphs not intersected.
graph%28600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2%2B2%2C+x-8%29

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Where f(x) and g(x) are equal.
f%28x%29=x%5E2%2B2+=+g%28x%29=x-8
x%5E2+%2B+2+=+x+-+8
x%5E2+-+x+%2B+10+=+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-1x%2B10+=+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=%28-1%29%5E2-4%2A1%2A10=-39.

The discriminant -39 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -39 is + or - sqrt%28+39%29+=+6.2449979983984.

The solution is , or
Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1%2Ax%2B10+%29

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The solutions are not real numbers, so the graphs do not intersect.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
They intersect where +f%28x%29+=+g%28x%29+
+x%5E2+%2B+2+=+x+-+8+
+x%5E2+-+x+=+-10+
There can be no intersection, because there is no way
the left side can equal the right
+x%5E2+-+x+ can only be positive
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Here's the graph:
+graph%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2+%2B+2%2C+x+-+8%29+