SOLUTION: {{{ log ( 3, ( x- 2) ) + log (3, ( x- 4) ) = 2 }}}

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Question 200476: +log+%28+3%2C+%28+x-+2%29+%29+%2B+log+%283%2C+%28+x-+4%29+%29+=+2+
Found 2 solutions by nerdybill, cordereal:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
Applying log rules:
+log+%28+3%2C+%28+x-+2%29+%29+%2B+log+%283%2C+%28+x-+4%29+%29+=+2+
+log+%28+3%2C+%28+x-+2%29%28+x-+4%29+%29+=+2+
+%28+x-+2%29%28+x-+4%29+=+3%5E2+
+x%5E2-+4x+-2x+%2B8+=+9+
+x%5E2-+6x+%2B8+=+9+
+x%5E2-+6x+-+1+=+0+
Solving for x using the quadratic equation yields:
x = {6.162, -0.162}
We can throw out the negative solution leaving:
x = 6.162
.
Details of quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-6x%2B-1+=+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=%28-6%29%5E2-4%2A1%2A-1=40.

Discriminant d=40 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+40+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+40+%29%29%2F2%5C1+=+6.16227766016838
x%5B2%5D+=+%28-%28-6%29-sqrt%28+40+%29%29%2F2%5C1+=+-0.16227766016838

Quadratic expression 1x%5E2%2B-6x%2B-1 can be factored:
1x%5E2%2B-6x%2B-1+=+1%28x-6.16227766016838%29%2A%28x--0.16227766016838%29
Again, the answer is: 6.16227766016838, -0.16227766016838. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-6%2Ax%2B-1+%29

Answer by cordereal(6) About Me  (Show Source):
You can put this solution on YOUR website!
6.16227766016838, -0.16227766016838