Solve for x e^(4x)+2e^(2x)-15=0

Problem

e^(4*x)+2*e^(2*x)−15=0

Solution

1. Substitute a new variable to transform the equation into a quadratic form. Let u=e^(2*x) Since e^(4*x)=(e^(2*x))^2 the equation becomes u^2+2*u−15=0

2. Factor the quadratic equation by finding two numbers that multiply to −15 and add to 2 These numbers are 5 and −3

(u+5)*(u−3)=0

3. Solve for u by setting each factor to zero.

u=−5

u=3

4. Back-substitute e^(2*x) for u and solve for x For the first case, e^(2*x)=−5 Since e^(2*x) must be positive for all real x there is no real solution for this case.

5. Apply the natural logarithm to the second case, e^(2*x)=3

ln(e^(2*x))=ln(3)

2*x=ln(3)

6. Isolate x by dividing both sides by 2

x=ln(3)/2

Final Answer

x=ln(3)/2

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