SOLUTION: {{{log10(x-6) + log10 (x-6) = 2 }}}

Algebra ->  Rational-functions -> SOLUTION: {{{log10(x-6) + log10 (x-6) = 2 }}}      Log On


   

Question 37545: log10%28x-6%29+%2B+log10+%28x-6%29+=+2+
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
log10(x-6) + log10 (x-6) = 2
log[(x-6)(x-6)]=2
log[x^2-12x+36]=2
x^2-12x+26=100
x^2-12x-74=
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B12x%2B-74+=+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=%2812%29%5E2-4%2A1%2A-74=440.

Discriminant d=440 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-12%2B-sqrt%28+440+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2812%29%2Bsqrt%28+440+%29%29%2F2%5C1+=+4.48808848170152
x%5B2%5D+=+%28-%2812%29-sqrt%28+440+%29%29%2F2%5C1+=+-16.4880884817015

Quadratic expression 1x%5E2%2B12x%2B-74 can be factored:
1x%5E2%2B12x%2B-74+=+1%28x-4.48808848170152%29%2A%28x--16.4880884817015%29
Again, the answer is: 4.48808848170152, -16.4880884817015. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B12%2Ax%2B-74+%29

Hope this helps.
Cheers,
Stan H.