SOLUTION: Good evening (Happy New Year to you all!!) Has somebody the patience to explain me step by step the solution of this exp.equation:? 4^(x-3)+2^(13-2x)=24 Results should be: x=9

Algebra ->  Equations -> SOLUTION: Good evening (Happy New Year to you all!!) Has somebody the patience to explain me step by step the solution of this exp.equation:? 4^(x-3)+2^(13-2x)=24 Results should be: x=9      Log On


   

Question 1012127: Good evening (Happy New Year to you all!!)
Has somebody the patience to explain me step by step the solution of
this exp.equation:? 4^(x-3)+2^(13-2x)=24
Results should be: x=9/2 or x=5
Thank you very much in advance
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
4^(x-3)+2^(13-2x)=24
4 = 2^2
2%5E%282x-6%29+%2B+2%5E%2813-2x%29+=+24
2%5E%282x%29%2F2%5E6+%2B+2%5E13%2F%282%5E%282x%29%29+=+24
2%5E%282x%29%2A2%5E%282x%29+%2B+2%5E13%2A2%5E6+=+24%2A2%5E6%2A2%5E%282x%29
2%5E%284x%29+-+24%2A64%2A2%5E%282x%29+%2B+2%5E19+=+0
Sub u for 2^2x
u%5E2+-+1536u+%2B+2%5E19+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-1536x%2B524288+=+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=%28-1536%29%5E2-4%2A1%2A524288=262144.

Discriminant d=262144 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1536%2B-sqrt%28+262144+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1536%29%2Bsqrt%28+262144+%29%29%2F2%5C1+=+1024
x%5B2%5D+=+%28-%28-1536%29-sqrt%28+262144+%29%29%2F2%5C1+=+512

Quadratic expression 1x%5E2%2B-1536x%2B524288 can be factored:
1x%5E2%2B-1536x%2B524288+=+%28x-1024%29%2A%28x-512%29
Again, the answer is: 1024, 512. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1536%2Ax%2B524288+%29

=================
2%5E%282x%29+=+1024+=+2%5E10
2x = 10
x = 5
--------------------------
2%5E%282x%29+=+512+=+2%5E9
x = 9/2