SOLUTION: log(x + 3) = 1 - log(x - 2) Solve for x

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Question 919205: log(x + 3) = 1 - log(x - 2) Solve for x
Found 2 solutions by ewatrrr, MathLover1:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
log(x + 3) = 1 - log(x - 2)
log(x + 3) + log(x - 2) = 1
log%2810%2C+%28x%2B3%29%28x-2%29%29+=+1
(x+3)(x-2) = 10
x^2 + x - 6 = 10
x^2 + x -16 = 0
x = 3.53112887414927
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B1x%2B-16+=+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=%281%29%5E2-4%2A1%2A-16=65.

Discriminant d=65 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+65+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+65+%29%29%2F2%5C1+=+3.53112887414927
x%5B2%5D+=+%28-%281%29-sqrt%28+65+%29%29%2F2%5C1+=+-4.53112887414927

Quadratic expression 1x%5E2%2B1x%2B-16 can be factored:
1x%5E2%2B1x%2B-16+=+1%28x-3.53112887414927%29%2A%28x--4.53112887414927%29
Again, the answer is: 3.53112887414927, -4.53112887414927. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B-16+%29

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!


log%28%28x+%2B+3%29%29+=+1+-+log%28%28x+-+2%29%29
log%28%28x+%2B+3%29%29%2B+log%28%28x+-+2%29%29=+1+
log%28%28x+%2B+3%29%28x+-+2%29%29=+1 ...........log%2810%2C10%29=1
log%28%28x+%2B+3%29%28x+-+2%29%29=log%2810%29 if log same, then
%28x+%2B+3%29%28x+-+2%29=+10
x%5E2-2x+%2B+3x+-+6=+10
x%5E2+%2B+x+-+6-10=0
x%5E2+%2B+x+-+16=0 .......use quadratic formula to solve for x

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A%28-16%29+%29%29%2F%282%2A1%29+
x+=+%28-1+%2B-+sqrt%28+1%2B64+%29%29%2F2+
x+=+%28-1+%2B-+sqrt%28+65+%29%29%2F2+
x+=+%281%2F2%29%28-1+%2B-+sqrt%28+65+%29%29+
solutions:
x+=+%281%2F2%29+%28-1-sqrt%2865%29%29+
x+=+%281%2F2+%29%28-1-8.06%29
x+=+%281%2F2%29+%28-9.06%29
x+=-4.53+

or
x+=+%281%2F2%29+%28sqrt%2865%29-1%29
x+=%28+1%2F2+%29%288.06-1%29
x+=+%281%2F2%29+%287.06%29
x+=3.53