SOLUTION: solve for x: log(base8)(2x+3) + log(base8)(x+1) = 1

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Question 274680: solve for x: log(base8)(2x+3) + log(base8)(x+1) = 1
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
log(base8)(2x+3) + log(base8)(x+1) = 1
.
log(base8)[(2x+3)(x+1)] = 1
.
8^log(base8)[(2x+3)(x+1)] = 8^1
.
(2x+3)(x+1) = 8
.
2x^2+2x+3x+3 = 8
.
2x^2+5x+3 = 8
.
2x^2+5x-5 = 0
.
Since we can't factor, we must resort to the quadratic equation which yields:
x = {0.766, -3.266}
Throw out the negative answer, it is an "extraneous" solution.
leaving:
x = 0.766
.
Details of the quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B5x%2B-5+=+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=%285%29%5E2-4%2A2%2A-5=65.

Discriminant d=65 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+65+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+65+%29%29%2F2%5C2+=+0.765564437074637
x%5B2%5D+=+%28-%285%29-sqrt%28+65+%29%29%2F2%5C2+=+-3.26556443707464

Quadratic expression 2x%5E2%2B5x%2B-5 can be factored:
2x%5E2%2B5x%2B-5+=+2%28x-0.765564437074637%29%2A%28x--3.26556443707464%29
Again, the answer is: 0.765564437074637, -3.26556443707464. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B5%2Ax%2B-5+%29