SOLUTION: a) solve the equation 7e^x + 3 = 2e^2x b) solve 1/2 |2-2x|<_4 and represent on the number line c) sole the equation x^4-5x^3+5x^2-5x-6=0

Algebra ->  Equations -> SOLUTION: a) solve the equation 7e^x + 3 = 2e^2x b) solve 1/2 |2-2x|<_4 and represent on the number line c) sole the equation x^4-5x^3+5x^2-5x-6=0       Log On


   

Question 1116196: a) solve the equation 7e^x + 3 = 2e^2x
b) solve 1/2 |2-2x|<_4 and represent on the number line
c) sole the equation x^4-5x^3+5x^2-5x-6=0
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
solve the equation 7e^x + 3 = 2e^2x
------------
2e^2x - 7e^x - 3 = 0
Sub for e^x
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-7x%2B-3+=+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=%28-7%29%5E2-4%2A2%2A-3=73.

Discriminant d=73 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--7%2B-sqrt%28+73+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-7%29%2Bsqrt%28+73+%29%29%2F2%5C2+=+3.88600093632938
x%5B2%5D+=+%28-%28-7%29-sqrt%28+73+%29%29%2F2%5C2+=+-0.386000936329383

Quadratic expression 2x%5E2%2B-7x%2B-3 can be factored:
2x%5E2%2B-7x%2B-3+=+%28x-3.88600093632938%29%2A%28x--0.386000936329383%29
Again, the answer is: 3.88600093632938, -0.386000936329383. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-7%2Ax%2B-3+%29

==================
e^x = ~ 3.886
x = ln(3.886) = 1.357
---------
Ignore the negative value.
===============
sole the equation x^4-5x^3+5x^2-5x-6=0
----
There are no integer zeroes.
Iterative or graphical methods can be used.