SOLUTION: Step-by-step solution of {{{ log(7, (3x-1)) + log(7, (2x+3)) = 2 }}} Note: Online math tools show that this doesn't have a solution. Would anyone be so kind of elaborating on th

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Step-by-step solution of {{{ log(7, (3x-1)) + log(7, (2x+3)) = 2 }}} Note: Online math tools show that this doesn't have a solution. Would anyone be so kind of elaborating on th      Log On


   

Question 1087461: Step-by-step solution of +log%287%2C+%283x-1%29%29+%2B+log%287%2C+%282x%2B3%29%29+=+2+
Note: Online math tools show that this doesn't have a solution. Would anyone be so kind of elaborating on this?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
+log%287%2C+%283x-1%29%29+%2B+log%287%2C+%282x%2B3%29%29+=+2+
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+log%287%2C%283x-1%29%2A%282x%2B3%29%29+=+2+=+log%287%2C7%5E2%29+=+log%287%2C49%29
%283x-1%29%2A%282x%2B3%29+=+49%29
6x%5E2+%2B+7x+-+3+=+49
6x%5E2+%2B+7x+-+52+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 6x%5E2%2B7x%2B-52+=+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=%287%29%5E2-4%2A6%2A-52=1297.

Discriminant d=1297 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-7%2B-sqrt%28+1297+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%287%29%2Bsqrt%28+1297+%29%29%2F2%5C6+=+2.41782385089485
x%5B2%5D+=+%28-%287%29-sqrt%28+1297+%29%29%2F2%5C6+=+-3.58449051756152

Quadratic expression 6x%5E2%2B7x%2B-52 can be factored:
6x%5E2%2B7x%2B-52+=+%28x-2.41782385089485%29%2A%28x--3.58449051756152%29
Again, the answer is: 2.41782385089485, -3.58449051756152. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B7%2Ax%2B-52+%29

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x = negative result --> log%287%2C%283x-1%29%29 = log of a negative number
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x =~ 2.41 is a valid result