SOLUTION: Please help me solve this equation: {{{log6 x + log6 (2x+1)=2 }}} Thank you very much.

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Question 339442: Please help me solve this equation:
log6+x+%2B+log6+%282x%2B1%29=2+
Thank you very much.
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
log6+x+%2B+log6+%282x%2B1%29=2+
log6+x%282x%2B1%29=2+
2x%5E2%2Bx+=+6%5E2+
2x%5E2%2Bx+=+36+
2x%5E2%2Bx-36+=+0+
x={4, -4.5}
The negative solution is extraneous, throw it out leaving:
x = 4
.
Details of quadratic to follow:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B1x%2B-36+=+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=%281%29%5E2-4%2A2%2A-36=289.

Discriminant d=289 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+289+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+289+%29%29%2F2%5C2+=+4
x%5B2%5D+=+%28-%281%29-sqrt%28+289+%29%29%2F2%5C2+=+-4.5

Quadratic expression 2x%5E2%2B1x%2B-36 can be factored:
2x%5E2%2B1x%2B-36+=+2%28x-4%29%2A%28x--4.5%29
Again, the answer is: 4, -4.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B1%2Ax%2B-36+%29