SOLUTION: Log(x-3) + log(2x + 1) = 2logx

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Question 192786: Log(x-3) + log(2x + 1) = 2logx
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
Log(x-3) + log(2x + 1) = 2logx
Log(x-3)(2x + 1) = logx^2
(x-3)(2x + 1) = x^2
2x^2 +x -6x - 3 = x^2
2x^2 - 5x - 3 = x^2
x^2 - 5x - 3 = 0
.
Using the quadratic equation to solve for x yields:
x = {5.541, -0.541}
We can toss out the negative solution leaving:
x = 5.541
.
Details of quadratic:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-5x%2B-3+=+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=%28-5%29%5E2-4%2A1%2A-3=37.

Discriminant d=37 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+37+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+37+%29%29%2F2%5C1+=+5.54138126514911
x%5B2%5D+=+%28-%28-5%29-sqrt%28+37+%29%29%2F2%5C1+=+-0.54138126514911

Quadratic expression 1x%5E2%2B-5x%2B-3 can be factored:
1x%5E2%2B-5x%2B-3+=+1%28x-5.54138126514911%29%2A%28x--0.54138126514911%29
Again, the answer is: 5.54138126514911, -0.54138126514911. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-5%2Ax%2B-3+%29