SOLUTION: Evaluate and get exact answer for: {{{ 10^x + 10^(-x) = 48 }}}

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Question 685172: Evaluate and get exact answer for:
+10%5Ex+%2B+10%5E%28-x%29+=+48+
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Evaluate and get exact answer for:
+10%5Ex+%2B+10%5E%28-x%29+=+48+
+10%5E%282x%29+%2B+1+=+48%2A10%5Ex+
+10%5E%282x%29+-+48%2A10%5Ex+%2B+1+=+0
u = 10^x
u%5E2+-+48u+%2B+1+=+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-48x%2B1+=+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=%28-48%29%5E2-4%2A1%2A1=2300.

Discriminant d=2300 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--48%2B-sqrt%28+2300+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-48%29%2Bsqrt%28+2300+%29%29%2F2%5C1+=+47.9791576165636
x%5B2%5D+=+%28-%28-48%29-sqrt%28+2300+%29%29%2F2%5C1+=+0.0208423834364027

Quadratic expression 1x%5E2%2B-48x%2B1 can be factored:
1x%5E2%2B-48x%2B1+=+%28x-47.9791576165636%29%2A%28x-0.0208423834364027%29
Again, the answer is: 47.9791576165636, 0.0208423834364027. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-48%2Ax%2B1+%29

===================
10^x = 24 + sqrt(575)
x = log(24 + sqrt(575))
x = log(24 - sqrt(575))