SOLUTION: Solve equation Logarithm {{{log(5,x^2+x+4)=2}}}

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Question 931627: Solve equation Logarithm
log%285%2Cx%5E2%2Bx%2B4%29=2
Found 2 solutions by ewatrrr, MathLover1:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
log%285%2Cx%5E2%2Bx%2B4%29=2
x^2+x+4 = 5^2
x^2 + x - 21 = 0
x =
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B1x%2B-21+=+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-21=85.

Discriminant d=85 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+85+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+85+%29%29%2F2%5C1+=+4.10977222864644
x%5B2%5D+=+%28-%281%29-sqrt%28+85+%29%29%2F2%5C1+=+-5.10977222864644

Quadratic expression 1x%5E2%2B1x%2B-21 can be factored:
1x%5E2%2B1x%2B-21+=+1%28x-4.10977222864644%29%2A%28x--5.10977222864644%29
Again, the answer is: 4.10977222864644, -5.10977222864644. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B-21+%29

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
log%285%2C%28x%5E2%2Bx%2B4%29%29=2....convert to log base 10
log%28%28x%5E2%2Bx%2B4%29%29%2Flog%285%29=2

log%28%28x%5E2%2Bx%2B4%29%29=2log%285%29
log%28%28x%5E2%2Bx%2B4%29%29=log%285%5E2%29....if log same, then
%28x%5E2%2Bx%2B4%29=5%5E2 is same too; solve for x
x%5E2%2Bx%2B4=25
x%5E2%2Bx%2B4-25=0
x%5E2%2Bx-21=0...use quadratic formula

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%28-21%29+%29%29%2F%282%2A1%29+
x+=+%28-1+%2B-+sqrt%28+1%2B84+%29%29%2F2+
x+=+%28-1+%2B-+sqrt%28+85+%29%29%2F2+
x+=+%28-1+%2B-+sqrt%28+85+%29%29%2F2+
exact solutions will be:
x+=+%28-1+%2B+sqrt%28+85+%29%29%2F2+
or
x+=+%28-1+-+sqrt%28+85+%29%29%2F2+

decimal solutions are:
x+=+%28-1+%2B+sqrt%28+85+%29%29%2F2+
x+=+%28-1+%2B+9.219544457292887%29%2F2+
x+=+8.219544457292887%2F2+
x+=4.109772228646444 exact decimal solution
x+=4.11 approximate
or
x+=+%28-1+-+sqrt%28+85+%29%29%2F2+
x+=+%28-1+-9.219544457292887%29%2F2+
x+=+-10.219544457292887%2F2+
x+=-5.10977222864644 exact decimal solution
x+=-5.11 approximate

+graph%28+600%2C+600%2C+-7%2C+7%2C+-5%2C+7%2C+log%285%2C%28x%5E2%2Bx%2B4%29%29%2C2%29+