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!!
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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(52800) (Show Source): You can put this solution on YOUR website!
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Find the sum of all solutions to e^{x^2}={e^{13x}}*{1/{e^{40}}}.
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You are given
= . .
It is the same as
= .
The last equality implies that
= .
Rewrite this quadratic equation as
= .
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 = , = .
The sum of the roots is 13.
Answer by Theo(13342) (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.
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