SOLUTION: log10(x+3)+log10(x-1)=1

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: log10(x+3)+log10(x-1)=1      Log On


   

Question 325591: log10(x+3)+log10(x-1)=1
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
+log%2810%2C%28x%2B3%29%29%2Blog%2810%2C%28x-1%29%29+=+1+
+log%2810%2C%28x%2B3%29%28x-1%29%29+=+1+
+%28x%2B3%29%28x-1%29+=+10%5E1+
+%28x%2B3%29%28x-1%29+=+10+
+x%5E2-x%2B3x-3+=+10+
+x%5E2%2B2x-3+=+10+
+x%5E2%2B2x-13+=+0+
Solve using the quadratic formula to get:
x = {2.741, -4.741}
The negative solution is extraneous leaving:
x = 2.741
.
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-13+=+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-13=56.

Discriminant d=56 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+56+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282%29%2Bsqrt%28+56+%29%29%2F2%5C1+=+2.74165738677394
x%5B2%5D+=+%28-%282%29-sqrt%28+56+%29%29%2F2%5C1+=+-4.74165738677394

Quadratic expression 1x%5E2%2B2x%2B-13 can be factored:
1x%5E2%2B2x%2B-13+=+1%28x-2.74165738677394%29%2A%28x--4.74165738677394%29
Again, the answer is: 2.74165738677394, -4.74165738677394. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B-13+%29