SOLUTION: I need help to see how to solve this problem. Find the sum of all solutions to e^{x^2}={e^{13x}}*{1/{e^{40}}}. Thanks for your help!!

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: I need help to see how to solve this problem. Find the sum of all solutions to e^{x^2}={e^{13x}}*{1/{e^{40}}}. Thanks for your help!!      Log On


   

Question 996404: I need help to see how to solve this problem. Find the sum of all solutions to e^{x^2}={e^{13x}}*{1/{e^{40}}}. Thanks for your help!!
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52792) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the sum of all solutions to e^{x^2}={e^{13x}}*{1/{e^{40}}}.
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You are given

e%5E%28x%5E2%29 = e%5E%2813x%29 . 1%2Fe%5E40.

It is the same as

e%5E%28x%5E2%29 = e%5E%2813x-40%29.

The last equality implies that

x%5E2 = 13x+-+40.

Rewrite this quadratic equation as

x%5E2+-+13x+%2B+40 = 0.

You can find its solution using quadratic formula,  or applying the Viete's theorem,  or by factoring.

In any case,  the roots of this quadratic equation are  x%5B1%5D = 5,   x%5B2%5D = 8.

The sum of the roots is  13.



Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your equation is:

e^(x^2) = e^(13x) * 1/e^40

this is equivalent to:

e^(x^2) = e^(13x) * e^(-40)

this is equivalent to:

e^(x^2) = e^(13x - 40)

this is true if and only if x^2 = 13x - 40

subtract 13x from both sides of this equation and add 40 to both sides of this equation to get:

x^2 - 13x + 40 = 0

factor this equation to get:

(x-8) * (x-5) = 0

set each of these factors equal to 0 and solve for x to get:

x = 8 or x = 5.

those are your solutions.