SOLUTION: log base 8 of (x) + log base 8 of (x+2)=3

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Question 572953: log base 8 of (x) + log base 8 of (x+2)=3
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
log base 8 of (x) + log base 8 of (x+2)=3
log base 8 of (x)(x+2)=3
(x)(x+2)=8^3
x^2+2x=512
x^2+2x-512 = 0
applying the quadratic formula results in:
x = {21.65, -23.65}
the negative solution is extraneous, throw it out leaving:
x = 21.65
.
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%2B2x%2B-512+=+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=%282%29%5E2-4%2A1%2A-512=2052.

Discriminant d=2052 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+2052+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282%29%2Bsqrt%28+2052+%29%29%2F2%5C1+=+21.6495033058122
x%5B2%5D+=+%28-%282%29-sqrt%28+2052+%29%29%2F2%5C1+=+-23.6495033058122

Quadratic expression 1x%5E2%2B2x%2B-512 can be factored:
1x%5E2%2B2x%2B-512+=+1%28x-21.6495033058122%29%2A%28x--23.6495033058122%29
Again, the answer is: 21.6495033058122, -23.6495033058122. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B-512+%29